Recurdyn Solver - Theoretical Manual

RReeccuurrDDyynn™™ // SSoollvveerr TThheeoorreettiiccaall MMaannuuaall

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This theoretical manual documents the theoretical background of the RecurDyn™ / Solver.

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Revision History First printed, April 2001 1st Revision, January 2002 2nd Revision, July 2002 3rd Revision, August 2002 4th Revision, September 2003 5th Revision, September 2005

RecurDyn™/SOLVER THEORETICAL MANUAL

TABLE OF CONTENTS

1. GENERALIZED RECURSIVE FORMULATION ………………… 1- 1

1.1. INTRODUCTION …………………………………………………………... 1- 1

1.2. RELATIVE COORDINATE KINEMATICS ……………………………….…. 1- 2

1.2.1 COORDINATE SYSTEMS ……………………………………………. 1- 2 1.2.2 RELATIVE KINEMATICS FOR A PAIR OF CONTIGUOUS BODIES …… 1- 4

1.3. GENERALIZATION OF THE VALOCITY RECURSIVE FORMULA …………... 1- 6

1.4. GENERALIZATION OF THE FORCE RECURSIVE FORMULA ……………… 1- 8

1.5. GRAPH REPRESENTATIONS OF MECHANICAL SYSTEMS ………………... 1- 9

1.6. EQUATIONS OF MOTION AND DAE SOLUTION METHOD …………….…. 1-10

1.7. GENERALIZED RECURSIVE FORMULAS …………………………………. 1-13

1.7.1 RECURSIVE FORMULA FOR ………………………………. xBX && = 1-14 1.7.2 RECURSIVE FORMULA FOR RM BOLD ………….…….. qq BxX )(= 1-14 1.7.3 RECURSIVE FORMULA FOR ………………………… q

Tq G)(Bg = 1-15

1.7.4 RECURSIVE FORMULA FOR AND ……….. qq x)B(X && = vv xBX )( && = 1-18

1.8. APPLICATION OF THE GENERALIZED RECURSIVE FORMULAS …………. 1-18

1.8.1 COMPUTATION OF THE RESIDUAL ……………………………… F 1-18 1.8.2 COMPUTATION OF THE JACOBIAN ……………………………… qF 1-19

1.9. NUMERICAL EXAMPLES ………………………………………..………... 1-20

1.9.1 A GOVERNOR MECHANISM …………..…………………………… 1-20 1.9.2 A MULTI-WHEELED VEHICLE ……….……………………….…… 1-22

1.10. CONCLUSIONS ………………………………………………..………… 1- 24

RecrDyn™ / Solver THEORETICAL MANUAL

REFERENCES ……………….…………………………………………….. 1- 25 APPENDIX A : RECURSIVE FORMULAS…………………………………… 1- 26

2. DECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION …………………………………….. 2- 1

2.1. INTRODUCTION …………………………………………………………... 2- 1 2.2. IMPLICIT NUMERICAL INTEGRATION FOR DIFFERENTIAL ALGEBRAIC EQUATIONS …….……………………………………………. 2- 2

2.3. A DECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION ……………………………………………….……………………………. 2- 4 2.4. NUMERICAL ALGORITHM ……………………………………………….. 2-7 2.5. NUMERICAL EXAMPLES …………………………………………………. 2-7

2.5.1 QUICK-RETURN MECHANISM …………………………………….. 2-7 2.5.2 AIR COMPRESSOR …………………………………………………. 2-10

2.6. CONCLUSIONS ………………………………………………..…………... 2-12 REFERENCES ...…………………………………………………………….. 2-13

3. FLEXIBLE MULTIBODY DYNAMICS USING A VIRTUAL BODY AND JOINT ………………………………………………………. 3- 1

3.1. INTRODUCTION …………………………………………………………... 3- 1 3.2. KINEMATICS TWO CONTIGUOUS FLEXIBLE BODIES ……………………. 3- 3

3.2.1. COORDINATE SYSTEMS AND VIRTUAL BODIES …………………… 3-3 3.2.2. JOINT CONSTRAINTS BETWEEN TWO RIGID BODIES …………….. 3-4 3.2.3. FLEXIBLE BODY JOINT CONSTRAINT BETWEEN A FLEXIBLE BODY AND A RIGID VIRTUAL BODY ………………………………………. 3-5

3.3. EQUATIONS OF MOTION …………………………………………………. 3- 9

3.3.1. COEFFICIENT MATRIX OF CONVENTIONAL AUGMENTED FORMULATION ……………………………………………………… 3-10 3.3.2. COEFFICIENT MATRIX OF PROPOSED AUGMENTED FORMULATION 3-11 3.3.3. NON-SINGULARITY OF AUGMENTED MASS MATRIX ……………… 3-12

3.4. COMPUTER IMPLEMENTATION AND DISCUSSIONS ………………………. 3-13

3.4.1. NUMERICAL ALGORITHM …………………………………………. 3-13 3.4.2. COMPARISON OF DIFFERENT IMPLEMENTATION METHODS ……… 3-14

3.5. NUMERICAL RESULTS …………………………………………………….. 3-17 3.5.1. FLEXIBLE SLIDER CRANK MECHANISM …………………………... 3-17 3.5.2. FLEXIBLE PENDULUM MECHANISM ………………………………. 3-20

3.6. SUMMARY AND CONCLUSIONS …………………………………………… 3-22 REFERENCES ...…………………………………………………………….. 3-23

4. GENERALIZED RECURSIVE FORMULATION FOR FLEXIBLE

MULTIBODY DYNAMICS ………………………………………… 4- 1

4.1. INTRODUCTION …………………………………………………………... 4- 1 4.2. RELATIVE COORDINATE KINEMATICS OF TWO CONTIGUOUS FLEXIBLE BODIES ……………………………………………………..….. 4- 3

4.2.1. COORDINATE SYSTEMS AND VIRTUAL BODIES ……………………. 4-3 4.2.2. RELATIVE KINEMATICS FOR A FLEXIBLE BODY JOINT …………… 4-5 4.2.3. RELATIVE KINEMATICS FOR A RIGID BODY JOINT ……………….. 4-7 4.2.4. GRAPH REPRESENTATIONS OF MECHANICAL SYSTEMS ………….. 4-8

4.3. FORWARD RECURSIVE FORMULAS ………………………………………. 4-10

4.3.1. GENERALIZATION OF THE VELOCITY RECURSIVE FORMULA ……. 4-10


4.3.2. RECURSIVE FORMULA FOR …………………………… qq )(BxX = 4-11 4.4. BACKWARD RECURSIVE FORMULAS ……………………………………... 4-13

4.4.1. GENERALIZATION OF THE FORCE RECURSIVE FORMULA ………… 4-13 4.4.2. Recursive Formula for ……………………………

kk qT

q )( GBg = 4-15

4.5. THE GOVERNING EQUATIONS OF SOLUTION …………………………….. 4-17 4.5.1. IMPLICIT INTEGRATION OF THE EQUATIONS OF MOTION ………… 4-17 4.5.2. APPLICATION OF THE GENERALIZED RECURSIVE FORMULAS …… 4-19

4.6. NUMERICAL RESULTS ……………………………………………………. 4-20 4.7. CONCLUSIONS ……………………………………………………………. 4-23 REFERENCES ...…………………………………………………………….. 4-25

5. RELATIVE NODAL METHOD FOR LARGE DEFORMATION PROBLEM

..………………………………………………………………….. 5-1

5.1. INTRODUCTION …………………………………………………………... 5-1 5.2. RELATIVE DEFORMATION KINEMATICS …………………………………. 5-2

5.2.1. GRAPH THEORETIC REPRESENTATION OF A STRUCTURE ………… 5-2 5.2.2. KINEMATIC DEFINITIONS ………………………………………….. 5-3

5.3. GOVERNING EQUATIONS OF EQUILIBRIUM ……………………………… 5-6 5.3.1. STRAIN ENERGY …………………………………………………… 5-6 5.3.2. EXTERNAL FORCE …………………………………………………. 5-7 5.3.3. CONSTRAINT ………………………………………………………. 5-8 5.3.4. EQUATIONS OF EQUILIBRIUM ……………………………………… 5-10

5.4. NUMERICAL ALGORITHM ………………………………………………… 5-11

5.5. NUMERICAL EXAMPLES ………………………………………………….. 5-12


6. DYNAMIC ANALYSIS OF HIGH-MOBILITY TRACKED VEHICLES

……………………………………………………….................... 6-1

6.1. INTRODUCTION …………………………………………………………... 6-1 6.2. HIGH-SPEED, HIGH-MOBILITY TRACKED VEHICLES …………………… 6-3

6.3. KINEMATIC RELATIONSHIPS AND EQUATIONS OF MOTION …………….. 6-6 6.4. A COMPLIANT TRACK MODEL ………………………………………….. 6-7

6.4.1. SINGLE PIN CONNECTION ………………………………………… 6-8 6.4.2. DOUBLE PIN CONNECTION ……………………………………….. 6-10

6.5. MEASUREMENT OF TRACK COMPLIANCE CHARACTERISTICS ………….. 6-12 6.6. CONTACT FORCES ……………………………………………………….. 6-15

6.6.1. INTERACTION BETWEEN TRACK AND ROAD WHEEL, IDLER, AND SUPPORT ROLLER …….…………………………………………… 6-15 6.6.2. TRACK CENTER GUIDE AND ROAD WHEEL INTERACTIONS ……… 6-16 6.6.3. INTERACTION BETWEEN SPROCKET TEETH AND TRACK LINK PINS 6-17 6.6.4. GROUND AND TRACK SHOE INTEGRATION …………………….. 6-18

6.7. METHOD OF NUMERICAL INTEGRATION …………………………………. 6-19

6.7.1. ACCURACY ANALYSIS ……………………………………………… 6-19 6.7.2. STABILITY ANALYSIS ………………………………………………. 6-20 6.7.3. IMPLEMENTATION OF A VARIABLE STEPPING ALGORITHM ………. 6-22

6.8. NUMERICAL RESULTS ……………………………………………………. 6-24



7. DYNAMIC TRACK TENSION OF HIGH MOBILITY TRACKED VEHICLES

…………………………………………………………………….

7-1

7.1. INTRODUCTION …………………………………………………………... 7-1 7.2. NUMERICAL MODEL OF A HIGH MOBILITY TRACKED VEHICLE ………... 7-3

7.3. INTERACTION GROUNDS …………………………………………………. 7-8 7.4. MEASUREMENT OF THE DYNAMIC TRACK ………………………………. 7-16 7.5. NUMERICAL INVESTIGATION OF DYNAMIC TRACK TENSION …………. 7-17 7.6. FUTURE WORK AND CONCLUSIONS ………………………………………. 7-22 REFERENCES ...…………………………………………………………….. 7-23

8. EFFICIENT CONTACT AND NONLINEAR DYNAMIC

MODELING FOR TRACKED VEHICLES ………………………….. 8-1

8.1. INTRODUCTION …………………………………………………………... 8-1

8.2. MULTIBODY TRACKED VEHICLE MODEL AND PARAMETER EXTRACTIONS ……………………………………………………………………………. 8-4

8.3. EFFICIENT CONTACT SEARCH ALGORITHM ……………………………. 8-5

8.3.1. ROAD WHEEL-TRACK LINK CONTACT …………………………… 8-5 8.3.2. SPROCKET-TRACK LINK CONTACT ………………………………... 8-6 8.3.3. GROUND-TRACK LINK CONTACT …………………………………. 8-7

8.4. EQUATIONS OF MOTION …………………………………………………. 8-10 8.5. EXTENDED BEKKER’S SOIL MODEL FOR MULTIBODY TRACK SYSTEM … 8-11 8.6. SUMMARY AND CONCLUSIONS …………………………………………… 8-15 REFERENCES ...…………………………………………………………….. 8-16

9. AN EFFICIENT CONTACT SEARCH ALGORITHM FOR

GENERAL MULTIBODY SYSTEM DYNAMICS ……………………. 9-1

9.1. INTRODUCTION …………………………………………………………... 9-1 9.2 KINEMATIC NOTATIONS OF A CONTACT PAIR ……………………………. 9-3

9.3. DIVISION OF THE CONTACT DOMAIN …………………………………….. 9-4 9.4. PRE-SEARCH ……………………………………………………………… 9-6

9.5. POST-SEARCH AND COMPLIANCE CONTACT FORCE …………………….. 9-7

9.6. KINEMATICS AND EQUATION OF MOTION FOR THE RECURSIVE FORMULAS ……………………………………………………………………………… 9-9 9.7. NUMERICAL INTEGRATION STRATEGY …………………………………… 9-12 9.8. NUMERICAL EXAMPLE ……………………………………………………. 9-13 9.9 CONCLUSIONS ……………………………………………………………... 9-15 REFERENCES ...…………………………………………………………….. 9-16

10. LINEARIZED EQUATIONS OF MOTION FOR MULTIBODY

SYSTEMS WITH CLOSE LOOPS …………………………………….. 10-1


10.1. INTRODUCTION ………………………………………………………….. 10-1 10.2. RELATIVE COORDINATE KINEMATICS …………………………………. 10-4

10.3. EQUATIONS OF MOTION ………………………………………………… 10-7 10.4. ELIMINATION OF LAGRANGE MULTIPLIERS AND LINEARIZATION OF THE EQUATIONS OF MOTION ……………………………………………. 10-8

10.5. NUMERICAL EXAMPLE…………………………………………………. 10-10

10.5.1. FOURBAR MECHANISM WITH A SPRING ………………………….. 10-10 10.5.2. CANTILEVER BEAM DRIVEN BY A MOTION ……………………… 10-13 10.5.3. A SPRING SYSTEM WITH 2 D.O.F ………………………………… 10-14 10.5.4. A CANTILEVER BEAM ……………………………………………. 10-15

10.9 CONCLUSIONS …………………………………………………………... 10-18 REFERENCES ...…………………………………………………………….. 10-19

11. NONLINEAR DYNAMIC MODELING OF

SILENT CHAIN DRIVE ………………………………………… 11-1

11.1. INTRODUCTION ………………………………………………………….. 11-1 11.2. MULTIBODY MODELING OF SILENT CHAIN DRIVE …………………….. 11-4

11.2.1. SPROCKET ………………………………………………………… 11-4 11.2.2. SILENT CHAIN LINK ………………………………………………. 11-4 11.2.3. TENSIONER AND CHAIN GUIDE ………………………………….. 11-6 11.2.4. EQUATIONS OF MOTION AND INTEGRATION …………………….. 11-6

11.3. CONTACT FORCE ANALYSIS …………………………………………….. 11-8

11.3.1. STRATEGE OF CONTACT SEARCH ………………………………… 11-8 11.3.2. LINE-ARC CONTACT ……………………………………………… 11-9 11.3.3. ARC-POINT CONTACT ……………………………………………. 11-12 11.3.4. ARC-ARC CONTACT ………………………………………………. 11-13

11.3.5. LINE-POINT CONTACT ……………………………………………. 11-13

11.3.6. CONTACT FORCE MODEL ………………………………………… 11-14 11.4. NUMERICAL STUDY OF AN AUTOMOTIVE SILENT CHAIN SYSTEM ……. 11-14 11.5. FUTURE WORK AND CONCLUSIONS ……………………………………. 11-18 REFERENCES ...…………………………………………………………….. 11-19

12. DYNAMIC ANALYSIS AND CONTACT MODELING FOR

TWO DIMENSIONAL MEDIA TRANSPORT SYSTEM ……………. 12-1

12.1. INTRODUCTION ………………………………………………………….. 12-1 12.2. TWO DIMENSIONAL FLEXIBLE MULTIBODY SHEET ……………………. 12-3 12.3. CONTACT FORCE ANALYSIS …………………………………………….. 12-5

12.3.1. KINEMATICS NOTATIONS …………………………………………. 12-6 12.3.2. SHEET AND ROLLER INTERACTIONS ……………………………... 12-7 12.3.3. ROLLERS INTERACTIONS ………………………………………… 12-9 12.3.4. SHEET AND ROLLER INTERACTIONS …………………………….. 12-9

12.4. EQUATIONS OF MOTION ………………………………………………… 12-12 12.5. NUMERICAL RESULTS …………………………………………………… 12-14 12.6. CONCLUSIONS …………………………………………………………… 12-16 REFERENCES ...…………………………………………………………….. 12-17

13. HYDRAULIC AUTO TENSIONER (HAT) FOR

BELT DRIVE SYSTEM ……………….………………………….. 13-1

13.1. INTRODUCTION ………………………………………………………….. 13-1


13.2. MULTIBODY SIMULATION MODEL …………………………..…………. 13-3 13.3. EQUATIONS OF MOTION ……………………………………………….. 13-4 13.4. HYDRAULIC FORCES …………………………………………………… 13-5

13.4.1. OIL FLOW RATES THROUGH THE CHECK VALVE …………………. 13-6 13.4.2. OIL FLOW RATES THROUGH THE LEAK

BETWEEN PLUNGER AND CYLINDER ……………………………... 13-7 13.5. CONTACK OF THE CHECK BALL …………..…………………………… 13-10 13.6. BELT DRIVE SYSTEM…………………………………………………….. 13-12 13.7. NUMERICAL RESULTS…………………………………………………… 13-13 13.8. CONCLUSIONS …………………………………………………………… 13-19 REFERENCES ...…………………………………………………………….. 13-20

14. DYNAMIC ANALYSIS OF CONTACTING SPUR

GEAR PAIR FOR FAST SYSTEM SIMULATION ……………….…. 14-1

14.1. INTRODUCTION ………………………………………………………….. 14-1 14.2. TOOTH PROFILE OF SPUR GEAR ……………………………………….. 14-2 14.3. EFFICIENT CONTACK SEARCH ALGORITHM AND

CONTACK FORCE MODEL ………………………………………….…… 14-4

14.3.1. ARC-ARC CONTACT ………………………………………………. 14-4 14.3.2. ARC-POINT CONTACT …………………………………………….. 14-8

14.4. KINEMATICS AND EQUATION OF MOTION FOR SYSTEM DYNAMICS

USING THE RECURSIVE FORMULAS …………..……………………… 14-9 14.5. NUMERICAL RESULTS…………………………………………………… 14-12


14.6. CONCLUSIONS …………………………………………………………… 14-17

REFERENCES ...…………………………………………………………….. 14-18

1

GENERALIZED RECURSIVE FORMULATION

1.1. INTRODUCTION

In Ref. 1, the equations of motion for the constrained mechanical systems were derived with respect to Cartesian coordinates. Then the equations were transformed into the corresponding ones that employ the relative coordinates by using the velocity transformation method. Since the virtual displacement and acceleration of the entire system were simultaneously substituted into the variational form of the equations of motion, the resulting equations of motion were compact. In spite of the compactness, they are not computationally efficient since the recursive nature of the relative kinematics was not exploited.

In Ref. 2, Hooker proposed a recursive formulation for the dynamic analysis of a satellite which has a tree topology. It was shown that the computational cost of the formulation increases only linearly with respect to the number of bodies. In Ref. 3, Featherstone proposed a recursive formulation to calculate the acceleration of robot arms using screw notation. These ideas were extended by using the variational vector calculus for constrained mechanical systems in Ref. 4.

Constrained mechanical systems are represented by differential equations of motion and algebraic constraint equations, which are often called differential algebraic equations (DAE). Several DAE solution methods using the BDF have been proposed in Refs. 5-7. In particular, the parameterization method treated the DAE as an ordinary differential equation (ODE) on the kinematic constraint manifolds of the system. The stability and convergence of the method were proved in Ref. 8. The present research employs this method, due to its mathematical soundness.

In Ref. 9, a recursive formulation was presented to obtain the Jacobian in the linearization of the equations of motion. Recursive formulas for each term in the equations of motion were directly derived, using the state vector notation.



Similar approach was taken in Ref. 8 to implement the implicit BDF integration with the relative coordinates. Since the recursive formulas were derived term by term, the resulting equations and algorithm became much complicated.

To avoid the complication involved in Ref. 8, the equations of motion are derived in a compact matrix form by using the velocity transformation method in the present study. Computational structure of the equations of motion in the joint space is carefully examined to classify all computational operations that can be done in a recursive way into several categories. The generalized recursive formula for each category of the computational operations is then developed and applied whenever such a category is encountered. Many common fact ors, which are not easily observed when they are derived term by term, can be observed among terms in the equations of motion. Furthermore, the matrix form of the equations makes it easy to debug and understand the program while computational efficiency is achieved by the recursive computational operation. A library of the generalized recursive formulas is developed to implement a dynamic analysis algorithm using the backward difference formula (BDF) and the relative generalized coordinate.

Section 2 introduces relative coordinate kinematics. Generalization of velocity and force recursive formulas is treated in Sections 3 and 4, respectively. Also, computational equivalence between the recursive method and velocity transformation method for a mechanical system is shown in Section 3. Section 5 presents a graph representation of mechanical systems. Section 6 presents the equations of motion and a solution method for the DAE. A library of the generalized recursive formulas are developed and applied in Sections 7 and 8. Numerical examples are given in Section 9. Conclusions are drawn in Section 10.

1.2. RELATIVE COORDINATE KINEMATICS

1.2.1 COORDINATE SYSTEMS

Orientation of a body in Fig. 2.1 is given as

1-3

[ hgfA =

=

333231

232221

131211

aaaaaaaaa

] (2-1)

where , , and h are unit vectors along the f g x′ , y′ , and axes, respectively. The

z′zyx ′−′−′ frame is the body reference frame and the

frame is the inertial reference frame. Z−YX −

Z

X

Y

rp

r

p

xy

zs

o

Fig. 2.1 Coordinate systems and a rigid body

Velocities and virtual displacements of point in the XO ZY −− frame are

defined as

(2-2a)

wr&

δπδr

(2-2b)

Their corresponding quantities in the zyx ′−′−′ frame are defined as



(2-3a)

≡

′′

=wArA

wr

Y T

T &&

(2-3b)

≡

′′

=Ζδπδ

πδδ

δ T

T

ArAr

1.2.2 RELATIVE KINEMATICS FOR A PAIR OF CONTIGUOUS BODIES

A pair of contiguous bodies is shown in Fig. 2.2. Body 1)(i − is assumed

to be an inboard body of body and the position of point is i iO

1)i(i1)i(i1)i(i1)(ii −−−− −++= sdsrr (2-4)

The angular velocity of body in its local reference frame, using Eq. 2-3a and defining , is

i

iT

1)(i1)i(i AAA −− =

1)i(i1)i(iT

1)i(i1)-(iT

1)i(ii −−−− ′+′=′ qHAwAw & (2-5)

where is determined by the axis of rotation. H′

zi

X

Z

Y

ri-1 ri

s(i-1)i si(i-1)

yi-1

xi-1

zi-1zi-1

xi-1

xi

yi

xi

zi

yiyi-1

d(i-1)i

oioi-1

Fig. 2.2 Kinematic relationship between two adjacent rigid bodies

1-5

Differentiation of Eq. 2-4, using Eq. 2-3a, yields

1)i(i'

1)i(i1)(i

'i

'1)i(ii

'1)(i

'1)i(i1)(i

'1)(i

'1)i(i1)(i

'1)(i1)(i

'ii

1)i(i)(

~~~

−−−

−−−−

−−−−−

−+

+−

−=

qdA

sAdA

sArArA

q &

&&

ωω

ω

(2-6)

where symbols with tildes denote skew symmetric matrices comprised of their vector elements that implement the vector product operation (Ref. 1) and

denotes the relative coordinate vector. Substituting of Eq. 2-5 and

multiplying both sides of Eq. 2-6 by yields

1)i(i−q'iω

TiA

1)i(i1)i(i1)i(iT

1)i(i1)i(i1)i(i1)i(iT

1)(i1)i(iT

1)-i(i1)i(i1)i(i1)i(i1)i(iT

1)(i1)i(iT

i

)~)((

)~~~(

1)i(i−−−−−−−

−−−−−−

−−

′′+′+

′′−′+′−

′=′

−qHAsAdA

AsAdsA

rAr

q &

&&

ω (2-7)

where iii

~ω′= AA& is used. Combining Eqs. 2-5 and 2-7 yields the recursive velocity equation for a pair of contiguous bodies.

1)i(i1)i2(i1)(i1)i1(ii −−−− += qBYBY & (2-8) where

′−′+′−

= −−−−−

−

−− I0

AsAdsIA0

0AB )~~~( 1)i(i

T1)i(i1)i(i1)i(i1)i(i

1)i(iT

1)i(iT

1)i1(i

′

′′+′

=

−

−−−−−

−

−−

−

1)i(i

1)i(i1)i(iT

1)i(i1)i(i1)i(i

1)i(iT

1)i(iT

1)i2(i

~)(1)i(i

HHAsAd

A00A

B q (2-9)

It is important to note that matrices and are functions of only

relative coordinates of the joint between bodies 1)i1(i−B 1)i2(i−B

(i 1)− and . As a

consequence, further differentiation of the matrices and in Eq.

i

1)i2−1)i1(i−B (iB



2-9 with respect to other than yields zero. This property plays a key role

in simplifying recursive formulas in Section 7. 1)i(i−q

1)i1(i− δ

2

Y

1)m1− Y

1)-(m

Similarly, the recursive virtual displacement relationship is obtained as follows.

(2-10) 1)i(i1)i2(i1)(iiδ −−− += δqBZBZ

1.3. GENERALIZATION OF THE VELOCITY RECURSIVE FORMULA

1

n-1

n

0

Fig. 3.1 A serial chain mechanism

Before proceeding to generalize the recursive velocity formula, the computational equivalence between the recursive method and the velocity transformation method is demonstrated using the mechanical system shown in Fig. 3.1. The Cartesian velocity is obtained by replacing by in Eq. 2-8.

m i m

1)m(m1)m2(m1)(m(mm −−− += qBBY & (3-1)

Substitutions of Eq. 2.8 for , , . . . , and yield Y 2)-(mY 0Y

1-7

1)m-(m1)m2-(m

1-m

1j1)j-(j1)j2-(j

j-m

1k1)1-mm)(-kk(

m

1k01)1-mm)(-kk(m

qB

qBB

YB

&

&

+

+

=

∏ ∏

∏

= =++−

=++−Y

(3-2)

Thus, the Cartesian velocity for all bodies is obtained as Y

qBY &= (3-3)

where is the collection of coefficients of and B 1)i(i−q&

[ ] T

1ncTT

2T

1T0 ×= nY,,Y,Y,YY K (3-4)

[ ] T1nr

T)1(

T12

T01

T0 ×−= nnq,,q,q,Yq &K&&& (3-5)

where and nc nr denote the numbers of the Cartesian and relative coordinates, respectively.

The Cartesian velocity , with a given , can be evaluated either

by using Eq. 3-3 or by using Eq. 2-8 with recursive numerical substitution of . Since both formulas give an identical result, and recursive numerical substitution is proven to be more efficient in Ref. 4, matrix multiplication with a given

will be evaluated by using Eq. 2-8.

ncR∈Y nrR∈q&

iY

qB &

q&

Since in Eq. 3-3 is an arbitrary vector in , Eqs. 2-8 and 3-3, which are

computationally equivalent, are actually valid for any vector such that

q& nrRnrR∈x

xBX &= (3-6)

and

1)i-(i1)i2-(i1)-(i1)i1-(ii xBXBX += (3-7)

where is the resulting vector of multiplication of and . As a ncR∈X B x



result, transformation of into is calculated by recursively applying Eq. 3.7 to achieve computational efficiency.

nrR∈x

Wδ

ncR∈Bx

G

ncR∈Q

QZΤ

*Tδ Qq=

+*

)!i(iQ

( + +1i1)2 Q

(

Tδq=

∑=

=1-n

0iδ

∑=

+

1-n

0i

Ti(iδq

T

=0

1.4. GENERALIZATION OF THE FORCE RECURSIVE FORMULA

It is often necessary to transform a vector in into a new vector

in . Such a transformation can be found in generalized force computation in the joint space with a known force in the Cartesian space. The virtual work done by a Cartesian force is

ncR

GBg T= nrR

δ= (4-1)

where must be kinematically admissible for all joints in Fig. 3.1. Substitution of

ZδqBZ δδ = into Eq. 4-1 yields

Tδ QBW (4-2)

where . Equation 4-2 can be written in a summation form as QBQ T* ≡

+T

1)i(iδ qW (4-3)

On the other hand, the symbolic substitution of the recursive virtual displacement relationship Eq. 2-10 into Eq. 4-1, along the chain in Fig. 3.1 starting from the body n toward inboard bodies, yields

)

)

++= 1iTi(i1)δ SBW (4-4)

where

2i2i2)11)(i(i1i

0

+++++ +≡ SQBS

S (4-5)

1-9

Equating the right sides of Eqs. 4-3 and 4-4, the following recursive formula for is obtained: *

Q

( ) 0 ...., 1,-ni,1i1i)21i(iT

1)i(i* =+≡ ++++ SQBQ (4-6)

where is defined in Eq. 4-5. 1i+S

Since is an arbitrary vector in , Eqs. 4-5 and 4-6 are valid for any

vector in . As a result, the matrix multiplication of is evaluated to achieve computational efficiency by

Q

G

ncRncR GBT

( )

( ) 0 ...., 1,i1i1i)11i(iT

i

n

1i1iT

1)2i(i1)i(i

=+≡

=

+=

+++

++++

SQBS

0S

SGBg

(4-7)

where is the result of . g GBT

1.5. GRAPH REPRESENTATIONS OF MECHANICAL SYSTEMS

In the previous section, a serial chain mechanism is considered to derive recursive formulas for and B where is a vector in and G

in . In general, a mechanical system may have various topological structures. To cope with the various topological structures, an automatic preprocessing is required for a general purposed program, which employs a relative coordinate formulation. The preprocessing identifies the topological structure of a constrained mechanical system to achieve computational efficiency. A graph theory was used to represent bodies and joints for mechanical systems in Refs. 1 and 4. A node and an edge in a graph represented a body and a joint, respectively. The preprocessing based on the graph theory yielded the path and distance matrices that are provided to automatically decide execution sequences

Bx GT x nrRncR



for a general purposed program. As an example, a governor mechanism and its graph representation are shown in Figs. 5.1 and 5.2.

4 3

2

87

6

5

1

U1

R2 R3

U2

S1 S2

T2

R1

T1

: Cut joint

Fig. 5.1 Governor mechanism

7

6

45

3

8

2

1

R1

T1

R3U2

R2

U1

T2

Cut JointCut Joint

Fig. 5.2 Graph representation of the governor mechanism

1.6. EQUATIONS OF MOTION AND DAE SOLUTION METHOD

The variational form of the Newton-Euler equations of motion for a constrained mechanism is

1-11

0)QλΦYMZ Τ

ΖT =−+&(δ (6-1)

where Zδ must be kinematically admissible for all joints except cut joints [1]. In the equation, and Φ λ , respectively, denote the cut joint constraint and the corresponding Lagrange multiplier. The mass matrix and the force vector

are defined as M

Q

( nbd21 ,,,diag MMMM L= ) (6-2)

′

=i

ii J0

0ImM (6-3)

[ ]Tnbd

T3

T2

T1 ,,,, QQQQQ L= (6-4)

~

′′′−′′′−′

=ωJωnrωmf

Q ~i

& (6-5)

where denotes the number of bodies, denotes the identity matrix, denotes the moment of inertia, denotes the external force, and denotes the external torque. Substituting the virtual displacement relationship into Eq. 6-1 yields

nbd I J′f ′ n′

0)QλΦYMBq Τ

ΖT =−+&(Tδ (6-6)

Since qδ is arbitrary, the following equations of motion are obtained:

0)QλΦYMBF ΤΖ

T =−+= &( (6-7)

The equations of motion, the constraint equations, q v=& , and constitute the following differential algebraic equations[8]:

av =&



0

avvqavqΦ

vqΦqΦ

)λa,vq(F

=

−−

)(

)(

)(

&

&

&&

&

t,,,t,,

t,t,,,

(6-8)

Application of 'tangent space method' in Ref. 7 to Eq. 6-8 yields the following

nonlinear system that must be solved at each time step:

0

βavU

βvqU

avqΦvqΦ

qΦ)λa,vq(F

pH =

++

++

)(

)(

)(

=

)β(

)β(

t,,,t,,

t,t,,,

)(

2n0nT0

1n0nT0

nnn

nnn

nn

nnnnn

nn

&&

& (6-9)

where [ ] TT

nTn

Tn

Tnn λ,a,v,qp =

0U

, , , and are determined by the coefficients of the BDF, and is an

0β 1β

nr2β

ncut)-(nr× such that the augmented

square matrix is nonsingular.

qΦUT

0

Applying the Newton's method to solve the nonlinear system in Eq. 6-9 yields

H∆pHp −= (6-10)

1,2,3,...i,in

1in =+=+ ∆ppp (6-11)

where

1-13

=

0UU000UU0ΦΦΦ00ΦΦ000Φ

FFFF

H

T0

avq

vq

q

qqqq

p

T00

T00

T0

ββ

&&&&&&

&& (6-12)

Since and are highly nonlinear functions of , , , and F Φ q v a λ , care

must be taken in deriving the non-zero expressions in , so that they can be

efficiently evaluated. pH

1.7. GENERALIZED RECURSIVE FORMULAS

Inspection of the residual and Jacobian matrix shows that types of

necessary recursive formulas are classified into Bx , , , ,

H pH

GBT xB& ( )qBx

( )qGBT , ( )qxB& , and ( )vxB& , where into G are arbitrary constant

vectors, and are relative coordinates. Formulas and were derived in Sections 3 and 4, and the formulas for the rest will be derived in this section. All recursive formulas are tabulated in Appendix A. Note that the recursive formulas are quite simple. This simplicity is achieved by exploiting the relative kinematics in the local reference frame instead of the global reference frame.

nrR∈x ncR

Bx

∈

q GBT

To derive the formulas systematically, bodies in a graph are divided into four disjoint sets (associated with a generalized coordinate ) as follows: kq

( ) kqqI havingjoint theofbody outboardbody outboradk =

coordinate dgeneralize its

as

( ) ( ) kk of bodies outboard all qIqII =

( ) ( )

=body inboard and base theincluding

, ofbody inboard theandbody base ebetween th bodies all kk

qIqIII



( ) ( ) ( ) ( ) kkkk ofset ary complement the qIIIqIIqIqIV UU=

For example, the body sets associated with q (relative coordinate between bodies 2 and 4) for the graph shown in Fig. 5.2 are obtained as follows:

24

( ) 4Body 24 =qI

( ) 7 and 6 Bodies 24 =qII ( ) 2 and 1 Bodies 24 =qIII ( ) 8 and 5 3, Bodies 24 =qIV

1.7.1 RECURSIVE FORMULA FOR xBX && =

Recursive formula for is easily obtained by differentiating Eq. 3-7. ncR∈xB&

1)i(i1)i2(i1i1)i1(i1i1)i1(ii −−−−−− ++= xBXBXBX &&&& (7-1)

This recursive formula can be applied to compute the Cartesian acceleration with known relative velocity and acceleration.

Y&

1.7.2 RECURSIVE FORMULA FOR RM BOLD qq BxX )(=

To obtain the recursive formula for , Eq. 3-7 is partially differentiated

with respect to . qBx)(

nr ..., 1, k,k =q

1)i(i1)2(i1-i1)i1-(i1i1)i1(ii kkk)()()()( −−−− ++= xBXBXBX qqqqk

(7-2)

Since matrices and depend only on the relative coordinates for

joint , their partial derivatives with respect to generalized coordinates

other than are zero. In other words, the partial derivatives are zero if

1)i1(i−B

1)i1

1)i2(i−B

1)i-(i

q (i− kq

1-15

does not belong to set . Therefore if body i is an element of set , Eq. 7-2 becomes

( )kqI II( )kq

(qIII

( )kq

i k) =X q

24

TG)(B

kk)()( 1i1)i1(ii qq XBX −−= (7-3)

If body i belongs to set ) ( )kk qIVU , is not affected by . As a result, Eq. 7-3 is further simplified as follows

iX kq

0X q =

k)( i (7-4)

If body i is an element of set , body is naturally its inboard body

and it belongs to set . Using Eq. 7-4, Eq. 7-2 becomes

( kqI ) 1)-(i

III

1)i(i1)2(i1i1)i1(i kk)()(( −−−− + xBXB qq (7-5)

This recursive formula can be applied to compute the partial derivative of the Cartesian velocity with respect to relative coordinates Y . For example, if

in Fig. 5.2, is shown in Fig. 7.1. q

24k qq = qY

1.7.3 RECURSIVE FORMULA FOR qT

q G)(Bg =

Recursive formula for is obtained by using the recursive formula in

Eq. 4-7. By replacing i by , Eq. 4-7 can be rewritten as q

-(i 1)

)(

)(

iiT

1)i1-(i1-i

iiT

1)i2-(i1)i-(i

SGBS

SGBg

+=

+= (7-6)

Taking partial derivative of Eq. 7-6 with respect to yields kq



(Y1)q24

= 0

(B671)q24 = 0,(B672)q24 = 0

(B461)q24 = 0,(B462)q24 = 0

(B241)q24 ,(B242)q24

(B121)q24 = 0,(B122)q24 = 0

(B231)q24 = 0,(B232)q24 = 0

(B351)q24 = 0,(B352)q24 = 0

(B281)q24 = 0,(B282)q24 = 0

(Y2)q24

= 0

(Y4)q24

=(B241)q24Y2

+(B242)q24q 24

(Y3)q24

= 0

(Y8)q24

= 0

(Y5)q24

= 0

.

(Y6)q24

= B 461 (Y4)q24

(Y7)q24

= B 671 (Y6)q24

Fig. 7.1 Computation Sequence for

24qY

kkk

kkk

)()()()()(

)()()()()(

iiT

1)i1(iiiT

1)i1(i1i

iiT

1)i2(iiiT

1)i2(i1)i(i

qqq

qqq

SGBSGBS

SGBSGBg

+++=

+++=

−−−

−−− (7-7)

Since is a constant vector, . If .ncR∈G 0G q =

k( ) ( ) ( kkk qIVqIIIqII UU∈ )i ,

and are not functions of . Therefore their partial derivatives

with respect to are zero. As a result, Eq. 7-7 can be simplified to 1)i1(i−B 1)i2−

kq(iB kq

kk

kk

)()()(

)()()(

iT

1)i1(i1i

iT

1)i2(i1)i(i

qq

qq

SBS

SBg

−−

−−

=

= (7-8)

Since for the tree end bodies, by the second equation 0S q =

k)( i 0S q =

k)( 1-i

1-17

of Eq. 7-8 for ( ) ( )kk qIVqII U∈i . Thus, for ( ) ( )kk qIVqII U∈i

)

, Eq. 7-8 becomes

g q− k)( 1)i(i

1)(i + 0S q =k

)( i

()(

()(T

1)i1(i1i

T(i1)i(i

k

k

BS

Bg

q

q

−−

−−

=

=

24k q=

= 0= 0

4

6

3

5

(g35)q24 = 0(S3)q24 = 0

(g23

(S2

)q24 = 0)q24 = 0

(g12)q24 = (B122)( S(S12)q24 =(B121)(S

2) q24

2) q24

2424 qq gG =

0= (7-9)

If , body ( ki qI∈ belongs to set ( )kqII , and . Thus, Eq. 7-7

becomes

)()

)()

i

i1)i2

k

k

S

S

q

q (7-10)

For example, if in Fig. 5.2, is shown in Fig. 7.2. q GB q24

T

1

(g67)q24

(S6)q24

2

8

(g28)q24 = 0(S2)q24 = 0

(g24)q24 = (B242)q24 S4

(S2)q24 =(B241)q24 S4

(g46)q24 = 0(S4)q24 = 0

7

Fig. 7.2 Computation Sequence for TB



1.7.4 RECURSIVE FORMULAS FOR AND qq x)B(X && = vv xBX )( && =

To obtain the recursive formula for and ( , Eq. 7-1 is partially

differentiated with respect to and for qx)B( &

k

vx)B&

nr ..., kq v 1, k = .

1)i(i1)i2(i1i1)i1(i1i1)i1(i

1i1)i1(i1i1)i1(ii

)()()(

)()()(

−−−−−−

−−−−

+++

+=

xBXBXB

XBXBX

qqq

qqq

kkk

kkk

&&&

&&& (7-11)

1)i(i1)i2(i1i1)i2(i

1i1)i1(i1i1)i1(ii

)()(

)()()(

−−−−

−−−−

++

+=

xBXB

XBXBX

vv

vvv

kk

kkk

&&

&&& (7-12)

The recursive formulas for and are obtained as in Appendix A

by following the similar steps taken in the previous sections. qx)B( & vx)B( &

1.8. APPLICATION OF THE GENERALIZED RECURSIVE FORMULAS

A library of the generalized recursive formulas is developed in Section 7. This section shows how the library can be utilized to compute the terms in and

in Eqs. 6.9 and 6.12. Inspection of and reveals that the residual

and partial derivatives of F , , , , , and Φ need to be

computed. Only and are presented in this section and the rest are

omitted for simplicity of the presentation.

H

pH

F

H

aFpH

Φ&q vF qΦ q q&&

F qF

1.8.1 COMPUTATION OF THE RESIDUAL F

The generalized force , and the Cartesian acceleration need to be

computed to obtain shown in Eq. 6-7. The term is obtained by applying the recursive formula in Eq. 7.1. The recursive formula with

in Eq 4.7 can be applied to evaluate in since

is a vector in

Q λΦZT Y&

F

)ncR .

Y&

GBT

nrR( T QλΦYMG Z −+= &

G

F

1-19

1.8.2 COMPUTATION OF THE JACOBIAN qFIn Eq. 6-7, differentiation of matrix with respect to vector results in a

three dimensional array. To avoid the complexity, Eq. 6-7 is differentiated with respect to a typical generalized coordinate . Thus,

B q

kq

nr.....,2,1,k,)(

)(

k

kk

TT

TT

=−++

−+=

qZ

Zqq

QλΦYMB

QλΦYMBF&

& (8-1)

Since the term can be easily expressed in terms of the Cartesian

coordinates, is obtained by applying the chain rule, as

)( T QλΦZ −

k)T

qZ Qλ −(Φ

kTT )()(

kBQλΦQλΦ ZZqZ −=− (8-2)

where BqZ=

∂∂ is used and denotes the kth column of the matrix . The

resulting equation for becomes

kB B

kqF

nr.....,2,1,k),)((

)(

kTT

TT

k

kk

=−++

−+=

BQλΦYMB

QλΦYMBF

ZZq

Zqq

&

& (8-3)

The first term in Eq. 8-3 can be obtained by applying the recursive formula for

, with , as explained in section 7.3. Collection of

, for all k, constitutes ( , which is equivalent to

. Matrix consists of columns which are

vectors in . Therefore, the application of , where is each column

of matrix , yields the numerical result of . Finally,

the second term in Eq. 8-3 is also obtained by applying , where

.

GB qkT

T( λΦZ −TT (( ZΦB

( MG =

)( T QλΦYMG Z −+= &

k

TT ))ZT( ZλΦ −

T)ZQλ −

)) kT BQλΦ ZZ −

) BQ Z

Qλ −ncRT( ZΦ

(k

Yq +&

BQλΦ ZZ )T −T)Z

GBT

Q nc

T (B

GTZλ

T)ZQΦ −

GBT



1.9. NUMERICAL EXAMPLES

1.9.1 A GOVERNOR MECHANISM

The mechanism shown in Fig. 9.1 consists of seven bodies, a spring-damper, five revolute joints, and a translational joint. The material properties and spring and damping constants of the system are shown in Table 9-1. The mechanism has redundant constraints that are removed by the Gaussian elimination with full pivoting. Consequently, it has only 2 degrees of freedom.

Dynamic analysis is carried out for 2 seconds with error tolerance of for the system. The Z acceleration of body 4 is drawn in Fig. 9.2. The result obtained by the other commercial program and that obtained by the proposed method are almost identical. The average step size, the numbers of residual function evaluations, and CPU time on SGI R3000 are shown in Table 9-2. The CPU time spent by the other commercial program is about 6 times larger than that by the proposed method. Note that the number of function evaluations of the proposed method is smaller than that of the other commercial program.

5103 −×

4 3

2

56

1

R4

R2 R3

R5

S1 S2

R1

T17

YX

Z

0.16

0.5

0.20.109

45o

Fig. 9.1 A governor mechanism

1-21

Table 9-1 Inertia properties of the governor mechanism and spring and damping

constants

Mass xI′ yI′ zI′ xyI′ yzI′ zxI′ Body 1 (Ground) not necessary

Body 2 200.0 25.0 50.0 25.0 0.0 0.0 0.0 Body 3 1.0 0.1 0.1 0.1 0.1 0.1 0.1 Body 4 1.0 0.1 0.1 0.1 0.1 0.1 0.1 Body 5 1.0 0.15 0.125 0.15 0.0 0.0 0.0 Body 6 0.1 0.1 0.1 0.1 0.0 0.0 0.0 Body 7 0.1 0.1 0.1 0.1 0.0 0.0 0.0

Spring constant 1000 Damping constant 30

Table 9-2 Integration output information

Program TOL Average step size No. fevals CPU time (sec) Other 5103 −× 2101.1 −× 748 41

Proposed 5103 −× 2102.1 −× 441 7

— PROPOSED … OTHER

Fig. 9.2 Z acceleration of Body 4



1.9.2 A MULTI-WHEELED VEHICLE

A vehicle example shown in Fig. 9.3 is chosen to show the practicality of the proposed method. The vehicle runs on a bump whose radius is 0.3048(m). The system consists of a chassis and twelve road wheels and arms. The material properties and spring and damping constants are shown in Table 9-3. The road wheel and arm are considered as a single body. As a result, the system has 18 degrees of freedom.

11.1 m/sec

0.3048m

Fig. 9.3 A multi-wheeled vehicle

Figure 9.4 shows the vertical acceleration of the chassis. It is shown that the

proposed method and the other commercial program yield almost identical results. The average stepsize, number of residual function evaluations, and CPU time on SGI R3000 are shown in Table 9-4. It can be shown that the proposed method performs much smaller number of residual function evaluations with larger stepsizes, and the CPU time by the proposed method is much shorter than that by the other commercial program. Since there is no closed chain in the system, the governing equations of motion are formulated as an ODE problem by the proposed method. On the other hand, the equations of motion by the other commercial program are formulated as an DAE problem. The DAE problem is generally more difficult to solve than the ODE problem. This general argument is supported by the numbers of function evaluation and average stepsize.

1-23

Table 9-3 Inertial properties of the vehicle mechanism and spring and damping constants

Mass xI′ yI′ zI′ xyI′ yzI′ zxI′ Body 1 (Ground) not necessary

Body 2 40773. 231800 60840 251700 - 234. -Body 3

~ Body 14 340.27 32.86 20.76 26.85 0.0 0.0 0.0

Spring constant 200000 Damping constant 40000

Table 9-4 Integration output information

Program TOL. Average step size No. fevals CPU time (sec) Other 410− 3104 −× 1359 330

Proposed 410− 3106.6 −× 1167 69

— OTHER … PROPOSED

Fig. 9.4 Vertical acceleration of the chassis



1.10. CONCLUSIONS

The recursive formulas are generalized in this research. The velocity transformation method is employed to transform the equations of motion from the Cartesian to the joint spaces. Computational structure of the equations of motion is examined to classify all necessary computational operations into several categories. The generalized recursive formula for each category is then applied whenever such a category of computation is encountered. Since the velocity transformation method yields the equations of motion in a compact form and computational efficiency is achieved by the generalized recursive formulas, the proposed method is not only easy to implement but also efficient. A dynamic analysis algorithm using the backward difference formula (BDF) and the relative generalized coordinate is implemented using the library of generalized recursive formulas developed in this research. Numerical studies showed that obtained solutions were numerically stable and computation time was reduced by an order of magnitude compared to a well-known commercial program.

1-25

REFERENCES

1. J. Wittenburg, Dynamics of Systems of Rigid Bodies, B. G. Teubner, Stuttgart, 1977.

2. Hooker, W., and Margulies, G., The Dynamical Attitude Equtation for an n-body

Satellite, Journal of the Astrnautical Science, Vol. 12, pp. 123-128, 1965.

3. R. Featherstone, The Calculation of Robot Dynamics Using Articulated-Body Inertias, Int.

J. Roboics Res., Vol 2 : 13-30, 1983.

4. D. S. Bae and Edward J. Haug, A Recursive Formulation for Constrained Mechanical

System Dynamics: Part II. Closed Loop Systems, Mech. Struct. and Machines, Vol. 15,

No. 4, pp. 481-506

5. Potra, F. A. and Petzold, L. R., ODAE Methods for the Numerical Solution of Euler-

Lagrange Equations. Applied Nume. Math., Vol. 10, pp. 397-413, 1992

6. Potra, F. A. and Rheinboldt, W. C., 1989, On the Numerical Solution of Euler-Lagrange

Equations, NATO Advanced Research Workshop on Real-Time Integration Methods for

Mechanical System Simulation, Snowbird, Utah, U. S. A..

7. Jeng Yen, Edward J. Haug, and Florian A. Potra, 1990, Numerical Method for

Constrained Equations of Motion in Mechanical Systems Dynamics, Technical Report R-

92, Center for Simulation and Design Optimization, Department of Mechanical

Engineering, and Department of Mathematics, The University of Iowa, Iowa City, Iowa.

8. Ming-Gong Lee and Edward J. Haug, 1992, Stability and Convergence for Difference

Approximations of Differential-Algebraic Equations of Mechanical System Dynamics,

Technical Report R-157, Center for Simulation and Design Optimization, Department of

Mechanical Engineering, and Department of Mathematics, The University of Iowa, Iowa

City, Iowa.

9. Lin, T. C. and Yae, K. H., 1990, Recursive Linearization of Multibody Dynamics and

Application to Control Design, Technical Report R-75, Center for Simulation and

Design Optimization, Department of Mechanical Engineering, and Department of

Mathematics, The University of Iowa, Iowa City, Iowa.



APPENDIX A : RECURSIVE FORMULAS

Recursive formulas

)(i kqI∈ )(i kqII∈

qq BxX )(= 1)i(i1)2(i

1i1)i1(ii

k

kk

)(

)()(

−−

−−

+

=

xB

XBX

q

qq kk 1i1)i1(ii qq )(XB)(X −−=

qT

q G)(Bg = i

Tiii

iT

iiii

k

k

( SBS

SBg

qq

qq

)()

)()(

1)1(1

2)1()1(

k

k

−−

−−

=

=

0S

0g

q

q

=

=

−

−

k

k

)(

)(

1i

1)i(i

qq x)B(X && =

1)i(i1)i2(i

1i1)i1(i

1i1)i1(ii

k

k

kk

)(

)(

)()(

−−

−−

−−

+

+

=

xB

XB

XBX

q

q

qq

&

&

&&

k

kk

)(

)()(

1i1)i1(i

1i1)i1(ii

q

qq

XB

XBX

−−

−−

+

=&

&&

vv xBX )( && = 1)i(i1)i2(i

1i1)i1(ii

k

kk

)(

)()(

−−

−−

+

=

xB

XBX

v

vv

&

&&

k

kk

)(

)()(

1i1)i2(i

1i1)i1(ii

v

vv

XB

XBX

−−

−−

+

=&

&&

Recursive formulas

)(i kqIII∈ )(i kqIV∈

qq BxX )(= 0X q =k

)( i 0X q =k

)( i

qq GBg )( T= kk

kk

)()(

)()(

iT

1)i1(i1i

iT

1)i2(i1)i(i

qq

qq

SBS

SBg

−−

−−

=

=

0S

0g

q

q

=

=

−

−

k

k

)(

)(

1i

1)i(i

qq xBX )( && = 0X q =k

)( i& 0X q =

k)( i

&

vv xBX )( && = 0X v =k

)( i& 0X v =

k)( i

& Recursive formulas

)(i kqI∈ or i or or )( kqII∈ )(i kqIII∈ )(i kqIV∈

BxX = 1)i(i1)i2(i1i1)i1(ii −−−− += xBXBX

GBg T= )(

0

)(

1i1iT

1)1i(ii

n

1i1iT

2)1i(i)1i(i

+++

++++

+=

=

+=

SGBS

S

SGBg

xBX && = i)1i(2i)1i(1i1i)1i(1i1i)1i(i −−−−−− ++= xBXBXBX &&&&

1-27



2

DECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL INTEGRATION

2.1. INTRODUCTION

The dynamic behavior of a constrained mechanical system is often represented by differential algebraic equations (DAEs)[1]. Solutions of DAEs are generally more difficult to obtain than those of ordinary differential equations (ODEs)[2]. To solve DAEs, a direct discretization method was proposed by Gear[3]. Since the solution obtained by Gear does not satisfy the velocity level constraints, consistent initial conditions cannot be obtained. It was found that the inconsistency often resulted in a poor local error estimation[4]. A series of stabilization methods[5-7] which employ either Lagrange multipliers or constraint violation penalty terms were followed.

Recently several solution methods[8], projecting the differential equations on the inflated constraint manifolds, have appeared. Two kinds of solution process are available. In the first solution process, the numerical integration is carried out first and the integrated variables are corrected so that the position level constraints, the velocity level constraints, and the acceleration level constraints are satisfied. Since the correction is made sequentially level-by-level, the size of system equations to be solved remains small. However, the integration stepsize can be excessively small for highly nonlinear or stiff problems due to a narrow stability region of the explicit method. In order to overcome this difficulty, the second solution process is developed. In the second solution process, the numerical integration formula, kinematic constraints and their derivatives, and equations of motion are solved simultaneously. Therefore, the size of the system equations to be solved becomes larger although the problem of excessive small step size is resolved. In addition to the problem of large size of the matrix equation, the condition of the matrix becomes poor as the stepsize gets smaller for discontinuous systems. The poor condition of the matrix often



results in large error in the solution of the matrix equation. In this paper, a decoupling solution method for the implicit numerical

integration method is proposed. This method is free from the problems of the poor matrix condition and the excessively small step size as well as the large matrix size.

In section 2, overdetermined DAEs for constrained mechanical systems are given. A decoupling solution method is given in section 3. In section 4, the numerical algorithm is provided. The numerical examples are given in section 5 to demonstrate the efficiency of the proposed method. Conclusions are drawn in section 6.

2.2. IMPLICIT NUMERICAL INTEGRATION FOR DIFFERENTIAL

ALGEBRAIC EQUATIONS

The equations of motion for a constrained mechanical system can be implicitly described as

0qv =− (2.1.a)

0λ)a,v,F(q, = (2.1.b) 0(q) =Φ (2.1.c)

where is the generalized coordinate vector in Euclidean space q nR , and λ

is the Lagrange multiplier vector for constraints in mR , Φ represents the position level constraint vector in mR , and its Jacobian is expressed

that is assumed to have full row-rank. Successive differentiations

of Eq. 2.1.c yield velocity and acceleration level constraints,

nq

×∈Φ mR

0υvv)(q, q =−= ΦΦ (2.2.a)

0γaa)v,(q, =−= qΦΦ& (2.2.b)

Equations 2.1 and 2.2 comprise a system of overdetermined differential algebraic

2-3

equations (ODAE). An algorithm based on backward differentiation formula (BDF) to solve the ODAE is given in Ref. 1 as follows:

0

ζqvU

ζvaU

Φ(q)

υvΦ

γaΦλ)a,v,F(q,

RU

RU

ΦΦΦ

F(x)

H(x)

T2

T1

q

q

2T2

1T1

=

−−

−−

−

−

=

=

20

10

0

0

bh

bh

bh

bh

&

&&

(2.3)

where ∑=

−=k

1i1ni

01 b

b1 vζ , ∑

=−=

k

1i1ni

02 b

b1 qζ

[λx =

iU

TqΦ

, k is the order of integration and

are the BDF coefficients. Here, and the columns of

constitute bases for the parameter space of the position

and velocity level constraints. are chosen so that has an inverse.

Therefore, the parameter space spanned by the columns of and the subspace

spanned by the columns of constitute the entire space

ib

iU

]q,v,a, TTTT

)2,1i()mn(n =∈ −×R

T

q

U

Φ

i

iUnR .

The number of equations and the number of unknowns in Eq. 2.3 are the same, so Eq. 2.3 can be solved. Newton's numerical method can be applied to obtain the solution . x

iii HxH x −=∆ (2.4.a) ii1i xxx ∆+=+ (2.4.b)

LU-decomposition of the matrix not only increases the computation time i

xH



but also produces an ill-conditioned matrix as h approaches zero [4]. In order to eliminate these problems, Eq. 2.4.a will be divided into several pieces to obtain

, , and q∆ v∆ a∆ λ∆ separately in the next section.

h ≡′

aF

a −

mR1τ ∈

v∆

2.3. A DECOUPLING SOLUTION METHOD FOR IMPLICIT NUMERICAL

INTEGRATION

Equation 2.4.a can be rewritten in detail as follows:

0xF∆λF∆aF∆vF∆qF λavq =++++ )( (3.1.a)

0xΦ∆qΦ∆vΦ∆aΦ qvq =+++ )(&&&&&& (3.1.b) 0xΦ∆qΦq =+ )( (3.1.c)

0xΦ∆qΦ∆vΦ qq =++ )(&& (3.1.d)

0xR∆v∆aU =′+−′ ))(( 1T1 hh (3.1.e)

0xR∆q∆vU =′+−′ ))(( 22 hhT (3.1.f)

where 0b

h

1U

. Equation 3.1.e can be rewritten in an equivalent inflated form

by choosing such that . as follows [7]: 0Tq

1a

T1 =− ΦFU

0τΦFUxR∆v∆ =′+′+′ −

11

1)( Tqa

Ti hhh (3.2)

where is a mass matrix and is generally nonsingular. The can be

singular if a parametric formulation is employed. If is singular, Eqs. 3.1

must be solved simultaneously to obtain

aF

aF

q∆ , v∆ , a∆ and λ∆ . The vector

is a new unknown variable. The a∆ is thus obtained from Eq. 3.2 in terms of as

1T1

a1 )(h1 τΦFxR∆v∆a q

−−−′

= (3.3)

2-5

Substituting Eq. 3.3 into Eq. 3.1.a yields

)()()(h 1 xRFxFτ∆λΦ∆vF

F∆qF a1Tq

avq +−=−+

′++ (3.4)

Equation 3.1.f can be rewritten in an equivalent inflated form by choosing

such that

2U

0ΦF

FU Tq

av =

′+

−1T2 h

as follows:

0τΦF

FxR∆q∆v Tq

av2 =

′+′+′+−′

−

2

1

)(h

hhh (3.5)

where h′

+ av

FF is assumed to be a nonsingular matrix and is a new

unknown variable. The solution process for the case of a singular matrix will be explained later in this section. Equation 3.5 can be solved for in terms of

as follows:

m2 Rτ ∈

v∆q∆

2

1

2 h)(

h1 τΦ

FFxR∆q∆v T

qa

v

−

′+−−

′= (3.6)

Substituting Eq. 3.6 into Eq. 3.4 and multiplying both sides of Eq. 3.4 by yields

2h′

32h RβΦ∆qK T

q* =′+ (3.7)

where

avq* FFFK +′+′≡ hh 2 (3.8.a)

21 ττ∆λβ −−≡ (3.8.b) )()h(h)(h)(h 21

223 xRFFxRFxFR ava +′′+′+′−≡ (3.8.c)



Equations 3.7 and 3.1.d are combined to obtain

−

=

′

)(h03

2 xΦR

β∆q

ΦΦK

q

Tq

*

(3.9)

Equation 3.9 is then solved for and . Note that is scaled by to make the coefficient matrix of Eq. 3.9 ill-conditioned, even as approaches to zero.

q∆ β β 2h′h′

Multiplying both sides of Eq. 3.5 by h′

+ av

FF yields

)(hh

1h 22 xR∆q

FFτΦ∆v

FF a

vTq

av −

′+

′=+

′+ (3.10)

Equations 3.10 and 3.1.c are combined to obtain

−−

−

′+

′=

′

+

∆qΦxΦ

xR∆qF

Fτ∆v

Φ

ΦF

F

q

av

q

Tq

av

)(

)(hh

1

0h 2

2

(3.11)

where has been obtained from Eq. 3.9. Equation 3.11 is solved for the

and . Multiplying both sides of Eq. 3.3 by yields

q∆

2

v∆

τ aF

∆

′−−=+ vxRFτΦ∆aF aa h

1)(11Tq (3.12)

Equations 3.12 and 3.1.b are combined to obtain

2-7

∆−−−

∆

′−−

=

qΦ∆vΦxΦ

vxRFτ∆a

ΦΦF

qv

a

q

Tqa

&&&&&& )(h1)(

01

1

(3.13)

Equation 3.13 is solved for a∆ and . Once , and are obtained, the

is evaluated from Eq. 3.8.b, as follows: 1τ β 1τ 2τ

∆λ

21 ττβ∆λ ++= (3.14)

Since is a mass matrix and is a tangent damping matrix, is generally not ill conditioned. If an ill-conditioned case is encountered, Eqs. 3.1 must be solved simultaneously to obtain

aF vF va FF h′+

q∆ , v∆ , a∆ and λ∆ . However, the

and aF vFhaF ′+ are rarely singular, so q∆ , v∆ , a∆ and λ∆ are obtained by using Eqs. 3.9, 3.11, 3.13, and 3.14 for most of practical problems.

2.4. NUMERICAL ALGORITHM

The DASSAL subroutine [4] is employed to integrate the system variables. Computational flow for the proposed DAE solution method is given in Fig. 1.(Page 2-7)


2.5.1 QUICK-RETURN MECHANISM

The quick-return mechanism as shown in Fig. 5.1 is mounted on a body

translating with respect to the ground. The system consists of 6 bodies, 2 translational joints, and 5 revolute joints. The system has two degrees of freedom if the redundant constraints are eliminated.

Dynamic analyses were performed for 1 sec with error tolerances of 10-4 and 10-6 by using the program developed in this paper and the other commercial program



Read initial conditions

Compute initial Accelerations and Lagrange multipliers from Eqs. 2.1.b and 2.2.b

t = t + h

Predict q v a, , , andλ

t > tout ?

YN

End

Y

F Fv

a+h'

N

Compute in Eq. 3.9 ∆q andβ

Compute in Eq. 3.11∆v andτ2

Compute

in Eqs. 3.1

∆ ∆ ∆ ∆λq, v, a, and

Update q v a, , , andλ

Convergence?N

Compute in Eq. 3.13∆a andτ1

Compute in Eq. 3.14∆λ

Faor is singular ? Y

Fig. 1 Flowchart for the proposed DAE solution method

2-9

Front v iew Side view

Body1Body2

Body3

Body5

Body6

Body4

Fig. 5.1 A quick-return mechanism

, which employs the implicit numerical integration with the BDF. The results are shown in Fig. 5.2. and the integration information is shown in Table 5.1.

—OTHER …PROPOSED

Fig. 5.2 Results of the quick-return mechanism



Table 5.1 Integration information for the quick return mechanism

Method Error Tolerance

No. Steps

No. Function

Evaluation

No. Jacobian

Evaluation

No. Newton IterationFailure

No. Integration

Failure

CPU Time

Proposed 1.0d-4 293 342 180 15 0 16 sec Other 1.0d-4 115 554 NA NA NA 34 sec

Proposed 1.0d-6 315 722 336 22 0 22 sec Other 1.0d-6 Failed to integrate.

(NA means Not Available)

Note that the other commercial program failed to integrate while the proposed method did not as the error tolerance became small (10-6). The integration failure was caused by the ill-conditioned Jacobian matrix

2.5.2 AIR COMPRESSOR

This system was modeled as four bodies, two revolute joints, two translational

joints, and 2 ball joints as shown in Fig. 5.3. The system has 1 degree of freedom if the redundant constraints are eliminated. Dynamic analyses were carried out for 1.0 sec with initial angular velocity. The proposed method and the other commercial program yielded identical results, as shown in Fig. 5.4. The system is conservative and the total energy should be constant. Figure 5.5 shows the total energy change during the integration. It is shown that the total energy obtained from the present program is numerically more stable than that obtained from the other commercial program. Thus, the other commercial program failed to integrate (while the proposed method did not) as the error tolerance became small. The integration information is also given in Table 5.2.

5 0 r a d / s e c

Fig. 5.3 An air compressor mechanism

2-11

Table 5.2 Integration information for the air compressor mechanism

Method Error Tolerance

No. Steps

No. Function

Evaluation

No. Jacobian

Evaluation

No. Newton IterationFailure

No. Integration

Failure

CPU Time

Proposed 1.0d-4 349 707 351 0 0 16 sec Other 1.0d-4 295 1185 NA NA NA 31 sec

Proposed 1.0d-6 529 1067 531 0 0 20 sec Other 1.0d-6 Failed to integrate.

— OTHER … PROPOSED

Fig. 5.4 Results for the air compressor



PROPOSEDOTHER

Fig. 5.5 Total energy comparison for the air compressor

2.6. CONCLUSIONS

A decoupling solution method for the implicit numerical integration is proposed in this paper. The size of the Jacobian matrix is significantly reduced by decoupling the iteration equations. The ill-conditioning problem of the implicit numerical integration is resolved in this method. Numerical study showed that the proposed method yields numerically more stable solution than the commercial program with smaller number of function evaluation.

2-13

REFERENCES

1. J. Yen, Constrained Equations of Motion in Multibody Dynamics as ODE's on Manifolds,

SIAM J. Numer. Anal., vol. 30 , pp. 553-568, (1993).

2. P. L. stedt and L. R.. Petzold, Numerical Solution of Nonlinear Differential Equations

with Algebraic Constraints I: Convergence Results for Backward Differentiation

Formulas, Math. Comp., vol. 46, pp. 491-516, (1986).

3. C. W. Gear, The Simultaneous Numerical Solution of Differential Algebraic Equations,

IEEE Trans. Circuit Theory, vol. 18, pp. 89-95, (1971).

4. K. E. Brenan, S. L. Campbell and L. R. Petzold, Numerical Solution of Initial-Value

Problems in Differential-Algebraic Equations, SIAM Press, (1995).

5. J. Baumgarte, Stabilization of Constraints and Integrals of Motion in Dynamical

Systems, Comput. Methods Appl. Mech. Engrg., vol. 1, pp. 1-16, (1972).

6. Javier Garcia de Jalon and Eduardo Bayo, Kinematic and Dynamic Simulation of

Multibody Systems, Springer-Verlag, (1993).

7. F. A. Potra, Implementation of Linear Multistep Methods for Solving Constrained

Equations of Motion, SIAM J. Numer. Anal., vol. 30, pp. 74-789, (1993).

8. Ming-Gong Lee and Edward J. Haug, Stability and Convergence for Difference

Approximations of Differential-Algebraic Equations of Mechanical System Dynamics,

Technical Report R-157, August, (1992).



3

FLEXIBLE MULTIBODY DYNAMICS USING A VIRTUAL BODY AND JOINT

3.1. INTRODUCTION

A rigid body in space is described by the position and orientation generalized coordinates with respect to the inertial reference frame. Contrast to implementation of a rigid body dynamic analysis program, it is generally complicated to implement a flexible body dynamic formulation and to expand it for a general purpose program, regardless of whatever formulation has been chosen. This is because the flexible body dynamic formulations handle additional generalized coordinates to these of the rigid body dynamics. One of the most tedious works involved with the implementation of the flexible body dynamics is to build a set of joint and force modules. Whenever a new force or joint module is developed for the rigid body dynamics, the corresponding module for the flexible body dynamics has to be formulated and programmed again. In order to avoid such a repetitive process, this investigation proposes a concept of virtual body and joint.

Shabana [1] presented a coordinate reduction method for multibody systems with flexible components. The local deformation of a flexible component was expressed in terms of the nodal coordinates and was then spanned by a set of mode shapes obtained from a mode analysis. Yoo and Haug [2] spanned the deformation by a set of static correction modes obtained by applying a unit force or unit displacement at a node where a large magnitude of force is expected during the dynamic analysis. Mani [3] used Ritz vectors in spanning the local deformation and the Ritz vectors were generated by spatially distributing the inertial and joint constraint forces on a flexible body. Gartia de Jalon et al [4] presented a fully Cartesian coordinate formulation for rigid multibody dynamics. This formalism was extended to the flexible body dynamics by Vukasovic et al [5]. Nonlinearity associated with an orientational transformation matrix was



relieved by defining all necessary vectors for the equations of motion and constraints as the generalized coordinates.

Several formulations have been recently developed for flexible body systems that undergo large deformation. Simo [6] had formulated the equations of motion for a flexible beam, based on the inertial reference frame. Since displacement of a point on the beam was directly measured from the inertial reference frame, the inertia terms become linear and uncoupled, while the strain energy related terms become nonlinear. Yoo and Ryan [7] proposed a mixed formulation of inertial and floating reference frames for a rotating beam. Axial deformation was measured from a deformed state of the rotating beam, while other deformations were measured from an undeformed state. Shabana [8,9] presented a non-incremental absolute coordinate formulation in which the global location coordinates and slopes were defined as the generalized coordinates. Since the finite rotation coordinates were not used as the generalized coordinates, the difficulties associated with the finite rotation were resolved.

Contrast to implementation of a rigid body dynamic analysis program, it is generally complicated to implement a flexible body dynamic formulation and to expand it for a general purpose program, regardless of whatever formulation has been chosen. This is because the flexible body dynamic formulations handle additional generalized coordinates to these of the rigid body dynamics. One of the most tedious works involved with the implementation of the flexible body dynamics is to build a set of joint and force modules. Whenever a new force or joint module is developed for the rigid body dynamics, the corresponding module for the flexible body dynamics has to be formulated and programmed again. In order to avoid such a repetitive process, this investigation proposes a concept of virtual body and joint. The kinematics of virtual body and joint is presented in Section 2. The equations of motion for a flexible body system are presented in Section 3. Computer implementation and its impact on a sparse oriented algorithm are explained in Section 4. Two flexible body systems are dynamically analyzed by using the proposed method to show its validity in section 5. Conclusions are drawn in Section 6.

3-3

3.2. KINEMATICS OF TWO CONTIGUOUS FLEXIBLE BODIES

3.2.1 COORDINATE SYSTEMS AND VIRTUAL BODIES

Figure 1 Two adjacent flexible bodies


Two flexible bodies connected by a joint and their reference frames are shown in Fig. 1. The frame is the body reference frame of flexible body and the frame is the inertial reference frame. Suppose there exists a

joint between the and frames, and a force applied at the

origin of the frame. Kinematic admissibility conditions among the reference frames can be divided into two categories. One is the admissibility conditions between the two joint frames and the other is the admissibility conditions among the frames within a flexible body. These two types of conditions have been mixed in formulating the kinematic joint constraints and generalized forces in the previous works. As a result, every joint and force modules in a flexible multibody code, such as ADAMS [10] and DAMS [11], has been developed separately for rigid and flexible bodies. This would take long time for computer implementation and prone to coding errors. Especially, flexible body programming requires much more effort than rigid body programming does due to complexity associated with flexibility generalized

iii ZYX ,,

ii YX 11 ,,iii ZYX 222 ,,

i

ZYX ,,iZ1

jjj ZYX 111 ,,


coordinates. In order to minimize the programming effort, a concept of the virtual body is introduced in this section. At every joint and force reference frames, a virtual rigid body, whose mass and moment of inertia are zero, is introduced.

Figure 2 Two adjacent flexible bodies and three virtual bodies

As an example, three rigid virtual bodies are introduced for two adjacent deformable bodies as shown in Fig. 2. This makes the flexible body has no joint or applied force and is subjected to only the kinematic admissibility conditions among its body frame and the virtual body frames. Therefore, the joint and force modules are developed only for rigid bodies and one flexible body joint is to be added in the joint module. The kinematic admissibility conditions for the flexible body joint are formulated in the following subsections.

3.2.2 JOINT CONSTRAINTS BETWEEN TWO RIGID BODIES

A joint has been represented by imposing condition of parallelism or

orthogonality on vectors attached to two adjacent rigid bodies. A library of such condition for rigid bodies has been well developed and becomes the primitives in building various joints [10, 11]. The conditions are formulated by using

3-5

geometric vectors that are defined within or between two joint reference frames. A joint reference frame does not generally coincide with the body reference frame. The body reference frame for a virtual body also serves as a joint reference frame in the proposed method. Therefore, the kinematic admissibility conditions for a joint connecting a virtual body is simplified and the number of non-zero entries of the constraint Jacobian is reduced.

3.2.3 FLEXIBLE BODY JOINT CONSTRAINT BETWEEN A FLEXIBLE BODY AND A

RIGID VIRTUAL BODY

Figure 3 Flexible body joint constraint between a flexible and a virtual body

Origin of the body reference frame for the virtual body in Fig. 3 can be

expressed as follows:

( if

iii

iiii

uuARuARr

++=

+=+

0

1

) (1)

where i

0u and ifu are the undeformed location vector and deformation vector of

a point on the body with respect to a body reference frame and is the orientation matrix of body reference frame. The deformation vector

iAifu at the



nodal position can be spanned by linear combination of a set of mode shapes [12] as

if

iR

if pu Φ= (2)

where is a modal matrix whose columns consist of the translational mode

shapes and is a modal coordinate vector.

iRΦ

ifp

Orientation of the virtual body is obtained as follows: 1+i

1,1 ++ = iiif

ii AAAA (3)

where is the relative orientation matrix induced by the rotational

deformation and is the orientation matrix between the reference frames of the flexible body and virtual body

ifA

1, +iiAi 1+i

if

in an undeformed state. If the Bryant angle (1-2-3) [13] is employed, the is expressed as follows: A

+−−−+

−=

iy

ix

iz

iy

ix

iz

ix

iz

iy

ix

iz

ix

iy

ix

iz

iy

ix

iz

ix

iz

iy

ix

iz

ix

iy

iz

iy

iz

iy

if

εεεεεεεεεεεεεεεεεεεεεεεε

εεεεε

coscossinsincossinsincossincossinsincossinsinsinsincoscoscossinsinsincos

sinsincoscoscosA (4)

If is infinitesimal, the matrix can be approximated as [ Ti

ziy

ix

i εεε=ε ] ifA

−−

−≈

11

1

ix

iy

ix

iz

iy

iz

if

εεεεεε

A (5)

The rotational deformation vector ε can be represented by linear

combination of rotational mode shapes of body as

i

i

if

ii pε θΦ= (6)

where is a modal matrix whose columns are composed of rotational mode iθΦ

3-7

shapes and is the vector of modal coordinate.

Ci

qδ

[

−=

000

ifp

Finally, kinematic constraints between two body frames of the flexible and virtual bodies can be obtained from Eqs. (1) and (3) as follows:

( ) 0uuARrC =+−−= + i

fiiiii

R 01 (7)

0hAAfhAAfhAAghAAggAAfgAAf

=

−

−

−

=++

++

++

1,1

1,1

1,1

iiif

TiTiT

iiif

TiTiT

iiif

TiTiT

θ (8)

where

[ ]

=

100010001

hgf (9)

Orthogonality conditions would have been used in deriving the orientational

constraints. However, the in Eq. (8) is employed in this research for simple implementation. Eqs. (7) and (8) yields algebraic constraint equations that describe the flexible joint between flexible body and virtual body . Taking variation of Eqs. (7) and (8) yields

iθC

i 1+i

( ) ( )( ) 0qCC

qCq

qq =

= δδ

θi

iRi

flexi (10)

where

[ ] TTiTiTif

TiTii 11 ++= πrpπR δδδδδ (11)

and the constraint Jacobian matrix ( )flex

iqC is obtained as

( ) ]

( )

−Φ−Φ−Φ

=

Φ−

+

+

+

1

1

1

ih

Tifh

Tih

T

ih

Tifh

Tih

T

ig

Tifg

Tig

T

i

iR

iiiR

Bf0BfBfBg0BgBgBf0BfBf

C

0IABIC

q

q

θ

θ

θ

θ

(12)



where

hgk

skew

skew

skew

skew

iTiik

iifk

iiTiik

iii

,,

)(

)(

)(

)(

11

1,

1

=

=

=

=

=

++

+

+

kAAB

kAB

AkAAB

uAB

(13)

and the vectors )( iskew u , , , and are the skew symmetric matrices of vectors,

)( 1kA +iskew )( 1, kA +iiskew )(kskewiu , , , and , respectively.

In order to obtain the acceleration level constraint, one can differentiate Eqs. (7) and (8) twice with respect to time to yield

kA 1+i A 1, +ii k k

( ) ( )( ) ( )

( )( )

+++++++++

Φ−×

=

=−=

4321

4321

4321

222

)(2)(

hT

hT

hT

hT

hT

hT

hT

hT

gT

gT

gT

gT

if

iR

iii

flexic

iiflex

iiflex

i

skewskew

HfHfHfHfHgHgHgHgHfHfHfHf

ωpAωωuA

QqqCqCqqq

&

&&&&

(14)

where the ω is the angular velocity with respect to the body reference frame and the generalized velocity vector is q&

[ ]TTiTiTif

TiTii 11 ++= ωrpωRq &&&& (15) and

( ) ( )

hgk

skew

skewskew

skewskewskew

skewskew

if

iiiif

if

ik

iiiTik

iiTiik

iTiiik

,,

)(

)()(

)()()(

)()(

1,4

1113

112

11

=

Φ×Φ=

=

=

=

+

+++

++

+

pkAApH

kωωAAH

ωkAAωH

kAAωωH

&& θθ

(16)

3-9

3.3. EQUATIONS OF MOTION

Even though the proposed method is applicable to a general system consisting

of many flexible bodies, a slider crank mechanism with one flexible body in Fig. 4(a) is used to clearly show the impact of the proposed method on the equations of motion. An equivalent virtual system, modeled by using the rigid virtual bodies proposed in this investigation, is shown in Fig. 4(b). The augmented equations of motion for the system is obtained by using the general form of equations of motion as [11]

++=

c

sveT

QQQQ

λq

0CCM

q

q && (16)

where is the mass matrix of the system. The vector consists of translational acceleration for rigid and flexible bodies, angular acceleration, and modal acceleration for the flexible body.

M q&&

FLEXIBLE BODYFLEXIBLE BODY

RIGID BODYRIGID BODY

CRAN

KCR

ANK

SLIDERSLIDER

COUPLERCOUPLER

1

2

3

Y

Z

X

P1P1

(a) Two rigid bodies and one flexible body



FLEXIBLE BODYFLEXIBLE BODY


VIRTUAL BODYVIRTUAL BODY

CRAN

KCR

ANK COUPLER

COUPLERSLIDERSLIDER

VIRTUAL BODYVIRTUAL BODY12

4

5

3

(b) Two rigid bodies, one flexible body and two virtual bodies

Figure 4 Slider crank mechanism with one flexible body

The is the vector of Lagrange multipliers and , , , and are the strain energy terms, velocity induced forces, externally applied forces, and the vector absorbs terns that are quadratic in the velocities, defined clearly by Shabana [11].

λ sQ vQ eQ cQ

cQ

3.3.1 COEFFICIENT MATRIX OF CONVENTIONAL AUGMENTED FORMULATION The mass matrix for the system in Fig. 4(b) is

=3

2

1

r

r

f

M0M

0MM (17)

where the mass matrices for virtual bodies, and are the mass matrix for

flexible body and for a rigid body, as fM rM

3-11

)3,2(,66

)6()6(

1

=

=

=

×

+×+

k

symmetric

k

krrk

r

nfnfffffr

r

rr

f

θθ

θ

θθθ

m00m

mmmmm

m

M

M

(18)

where is the number of modal coordinates. The constraint Jacobian matrix

of the slider crank mechanism with flexible crank is nf

C)( qC

( )( )( )( )

=

int30

int23

,12

,01

)(

jo

jo

cflex

cflex

c

q

q

q

q

q

CCCC

C (19)

where, ( )

cflex,qC is the constraint Jacobian matrix of the flexible joint obtained

by the conventional method[11].

3.3.2 COEFFICIENT MATRIX OF PROPOSED AUGMENTED FORMULATION

The mass matrix for the system in Figure 4b is

=

5

4

3

2

1

r

r

v

f

v

MM0

M0M

M

M (20)

where the mass matrix for virtual body, , the mass matrix for flexible body, vM



fM , and the mass matrix for rigid body, are rM

symmetric

m

,66×

int

int

int

,

,

int

p

p

[ ] 3,1,66 == × kv 0M

)6()6(

2

nfnfffffr

r

rr

f

+×+

=mmmm

mM

θ

θθθ (21)

)5,4( =

= kk

krrk

rθθm0

0mM

The proposed constraint Jacobian matrix ( )

pqC of the slider crank

mechanism with flexible crank is

( )

( )( )( )( )( )( )

=

50

45

34

23

12

01

jo

jo

jo

flex

flex

jo

p

q

q

q

q

q

q

q

CCCCCC

C (22)

where ( )

pflex,qC is the constraint Jacobian matrix of the flexible body joint

obtained by the proposed method. As shown in Eq. (22), the constraint Jacobian matrix can be clearly divided into flexible and rigid body joint modules by introducing rigid virtual bodies.

3.3.3 NON-SINGULARITY OF AUGMENTED MASS MATRIX

If the constraint Jacobian matrix C has a full row rank, the coefficient

matrix of Eq. (16) is non-singular, which can be proved by showing that the following equations have only trivial solutions under the same assumption.

q

3-13

( ) 0yCy q =+ 31 N

TNM (23)

( ) 0yCq =3VT (24)

( ) ( ) 0yCyC qq =+ 21 VN (25)

where , NM ( )

NqC , and ( )VqC are the mass matrix of non-virtual body, the

Jacobian of Eqs. (23) and (24) after pre-multiplying by Eq. (23) and by Eq. (24) yields

T1y T

2y

( ) ( )( ) 0yMyyCyCyyMy qq ==++ 1121311 NT

VNT

NT (26)

where Eq. (25) is used. As a result, . Eqs. (23) and (24) reduces to 0y =1

0yCq =3

T (27)

Since the has full row rank, must be zero. Substituting into Eq.

(25) yields qC 3y 0y =1

( ) 0yCq =2V (28)

Since rank of ( )

VqC

y =3

is the same as the size of , must be zero. Since

, , are only solutions of Eqs. (23), (24), and (25), their coefficient matrix is non-singular.

2y 2y

0y =1 0y =2 0

3.4. COMPUTER IMPLEMENTATION AND DISCUSSIONS

In previous sections the flexible equations of motion and kinematic constraints using virtual body techniques are presented. In this section, the computer implementation methods for the equations developed in section 2 and 3 are illustrated.

3.4.1 NUMERICAL ALGORITHM


A general purpose program for the dynamic analysis of mechanical systems


can be implemented in many different ways, depending on the DAE solution method employed. The generalized coordinate partitioning method [12] is employed in this investigation and the proposed program structure is shown in Fig. 5. Note that there exist joint and force modules only for rigid bodies, and one flexible body joint is added in the joint library. Those modules can handle any system consisting of rigid bodies as well as flexible bodies.

POSITION ANALYSIS

VELOCITY ANALYSIS

ACCELERATION ANALYSIS

FORCE ANALYSIS JOINT MODULES FOR RIGID BODY

FLEXIBLE BODY JOINT

JOINT MODULES

GENERATE JOINT EQUATIONS

FORCE MODULES FOR RIGID BODYT = T_END

NO

YES

END

START

T = T + STEPSIZE

MAIN PROGRAMMAIN PROGRAM MODULE LIBRARYMODULE LIBRARY

Figure 5 A program structure for proposed flexible multibody dynamics

3.4.2 COMPARISON OF DIFFERENT IMPLEMENTATION METHODS

The joint and force modules must be expanded whenever a user group of the flexible body dynamics code demands a special type of joint or force element. Since the proposed implementation method for a flexible body dynamics code reuses all joints and force modules for the rigid body, only necessary modules to be added to a rigid body dynamics code are the flexible body joint and the equations of motion for a flexible body. As a result, the proposed method is not only easy to implement but also to maintain, because the proposed method eliminates the additional programming effort for the flexible body modules when an expansion of the joint or force library is required.

However, there are some computational overheads, because extra bodies and joints must be introduced to a flexible body system if the proposed method is employed. It is very difficult to analyze the computational overheads for general

3-15

rigid and flexible multibody systems, because various models and flexible body dynamics theories may end up with various situations. In order to simplify the presentation, the slider crank mechanism in section 3 is reconsidered in this section.

Numerical experiments with the Cartesian coordinate formulation [12] showed that more than 70% of the total computation time is consumed in the Gaussian elimination of matrices arising from various equations. Direct Gaussian elimination of Eq. (16) would require a number of arithmetic operations proportional to approximately cube of the matrix size. However, the number of arithmetic operations for a sparse solver such as the Harwell Library [14] is increased only linearly to the number of non-zero entries if the structure of the non-zero entries is exploited. A sparse solver reduces the number of operations by minimizing the number of fill-ins and performing the Gaussian elimination only on the non-zero entries and fill-ins. Therefore, it is important to add the new non-zero entries so that overall non-zero structure of the resulting matrix is not disturbed and is well suited for minimization of the fill-ins. The structures of the non-zeros are shown in Eqs. (20) and (21), respectively. No non-zero entry in the mass matrix of the proposed method is added, because the mass and moment of inertia of the virtual body are zero. Total numbers of non-zero entries of Eq. (16) are shown in Table 1. Note that redundant constraints are eliminated and coincidence of the virtual body and joint reference frames is utilized in reducing the number of non-zeros. Since the new non-zero entries in Eq. (16) are scattered around the existing ones, the overall structure of the non-zeros is not disturbed and a similar reordering sequence in sparse Gaussian elimination to the original reordering sequence in a sparse linear solver can be used. As a result, expected computation time increment with the proposed method would be about 50% for the slider crank mechanism, when a sparse solver is employed.

Table 1 Number of non-zero entries for the slider-crank mechanism

Implementation Methods No. of non-zero entries Conventional 122+10×nmode

Propose 188+12×nmode * nmode: the number of mode shapes



The number of non-zeros for a most frequently used joints such as, revolute

joint, spherical joint and translational joint, are also given in Table 2. It can be easily shown that the percent ratio of the computation time would become smaller if the number of flexible bodies in a system is small, which is true in many cases. However, the computation time may be increased significantly for a flexible body system which has many joints and force elements, because the number of virtual bodies in such a system is large.

Table 2 Number of non-zero entries for the slider-crank mechanism

Joint Increment of non-zero entries

Revolute joint (33 + nomde) ×nvirtualr

Spherical joint (33 + 3×nomde)×nvirtuals

Translational joint (33 + nomde) ×nvirtualt

* nvirtualr : the number of virtual bodies which are connected with revolute

joint

* nvirtuals : the number of virtual bodies which are connected with spherical

joint

* nvirtualt : the number of virtual bodies which are connected with

translational joint

Another way of implementing the virtual body concept is to mix the proposed

implementation method with the conventional one. The conventional method may be used to implement the frequently used joint and force modules such as the revolute and translational joints and an applied force at a point. Meanwhile, the proposed method may be used to implement the less frequently used joint and force modules such as an universal joint or a planar joint. This implementation method will improve both the computational overhead as well as the coding convenience. This mixed formulation can be very effective if a set of basic joint and force modules have already been developed and more modules for the flexible bodies need to be added.

3-17

3.5. NUMERICAL RESULTS

Dynamic analysis of a flexible slider crank mechanism and a flexible pendulum mechanism is presented in order to validate the results from the proposed method. The examples are solved by using both the proposed method and the nonlinear approach developed by Simo [6].

3.5.1 FLEXIBLE SLIDER CRANK MECHANISM

The system consists of two rigid bodies and one flexible body, as shown in

Fig. 4. Length, cross sectional area, and area moment of inertia of the elastic crank are 0.4 m, 0.0018m2, and 1.215 ×10-4m4, respectively. The crank is modeled by using 10 two-dimensional elastic beam elements of equal lengths. The material mass density of the beam is 5540.0 kg/m3 and its Young's modulus is 1.0×109 N/m2. Vibration analysis of the crank is carried out with fixed-free boundary condition and the resulting mode shapes are shown in Fig. 6.

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

X(M)

MAG

NIT

UD

E.

1st mode

2nd mode

3rd mode

4th mode

Figure 6 Mode shapes of the crank

Four mode shapes are selected to span the deformation of the crank. As a result,



the system has 5 degrees of freedom. Dynamic analysis using the generalized coordinate partitioning method is performed for 5 sec under the constant acceleration condition of the joint between the ground and the body 1. The acceleration, displacement, and relative deformation of the pin joint connecting the crank and the coupler both from the proposed method and the nonlinear approach [6] are shown in Figs. 7, 8, and 9, respectively. Note that since the results from both models are almost identical as shown in these figures, the proposed implementation methods using rigid virtual body can be validated.

-30

-20

-10

0

10

20

30

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

TIME (SEC)

Y (

M/S

EC

^2).

.

NONLINEAR

PROPOSED

Figure 7 Y Acceleration of P1

3-19

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

TIME (SEC)

Y (

M)

NONLINEAR

PROPOSED

Figure 8 Y Displacement of P1

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

TIME (SEC)

Y (

M)

NONLINEAR

PROPOSED

Figure 9 Deformation of P1



3.5.2 FLEXIBLE PENDULUM MECHANISM

BEAMBEAM

(FLEXIBLE BODY)(FLEXIBLE BODY)


REVOLUTEREVOLUTE JOINT JOINT

GRAVITYGRAVITY

X

Y

Figure 10 Simple flexible pendulum model

The pendulum body shown in Fig. 10 is modeled with 10 beam elements

having a length of 0.4m, a cross sectional area of 0.0018m2, and a mass of 3.9888kg. Dynamic analysis is performed for 1 sec under the free falling condition. Mode shapes of the pendulum are obtained by ANSYS[15] with the simply supported-free(pin-free) boundary condition. Mode Shapes of the pendulum are shown in Fig. 11. The acceleration and relative transverse deformation of the tip point both from the proposed method and the nonlinear approach [6] are shown in Figs. 12, and 13, respectively. It is clear from these results that the proposed method and nonlinear approach are in good agreement, accordingly.

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

X(M)

MA

G

1st mode

2nd mode

3rd mode

4th mode

3-21

Figure 11 Mode Shapes of the pendulum

-30

-20

-10

0

10

20

30

40

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

TIME (SEC)

Y (

M/S

EC

^2).

.

NONLINEAR

PROPOSED

Figure 12 Y Acceleration of beam tip

-0.0015

-0.0010

-0.0005

0.0000

0.0005

0.0010

0.0015

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

TIME (SEC)

Y (

M)

NONLINEAR

PROPOSED

Figure 13 Deformation of beam tip



3.6. SUMMARY AND CONCLUSIONS

An implementation method is proposed for general purpose rigid and flexible multibody dynamics with the Cartesian coordinate formulation. A concept of the virtual body and joint is introduced to make a flexible body free from all kinematic admissibility conditions except these from the virtual-flexible body joint. This eliminates extra programming efforts for the flexible body whenever a joint or force module is added to a general purpose dynamic analysis program. The computational overhead of the proposed method is turned out to be moderate if a sparse solver is employed, while implementation convenience is dramatically improved. A flexible slider crank mechanism and a simple pendulum are analyzed and the results are validated against these from a nonlinear approach.

3-23

REFERENCES

1. A. A. Shabana, "Substructure Synthesis Methods for Dynamic Analysis of Multibody

Systems", Computers & Structures, Vol. 20. No. 4, pp 737-744, 1985

2. W. S. Yoo, and E. J. Haug, "Dynamics of Flexible Mechanical Systems Using Vibration

and Static Correction Modes", Journal of Mechanisms, and Transmissions, and

Automation in Design, 1985

3. H. T. Wu, and N. K. Mani, "Modeling of Flexible Bodies for Multibody Dynamic Systems

Using Ritz Vectors", Journal of Mechanical Design, Vol. 116, pp. 437-444, 1994.

4. J. Garcia de Jalon, J. Unda, and A. Avello, "Natural Coordinates for the Computer

Analysis of Three-Dimensional Multibody Systems", Computer Methods in Applied

Mechanics and Engineering, Vol. 56, pp. 309-327, 1985.

5. N. Vukasovic, J. T. Celigueta, J. Garcia de Jalon, and E. Bayo, "Flexible Multibody

Dynamics Based on a Fully cartesian System of Support Coordinates", Journal of

Mechanical Design, Vol. 115, pp. 294-299, 1993.

6. J. C. Simo, and L. Vu-Quoc, "On the Dynamics of Flexible Beams Under Large Overall

Motions-The Plane Case: Part I", Journal of Applied Mechanics, Vol. 53, pp. 849-854,

1986.

7. H. H. Yoo, R. R. Rion, and R. A. Scott, "Dynamics of Flexible Beams Undergoing

Overall Motions", Journal of Sound and Vibration, Vol. 181, pp. 261-278, 1994

8. A. A. Shabana, A. P. Christensen, "Three Dimensional Absolute Nodal Coordinate

Formulation : Plate Problem", International Journal for Numerical Methods in

Engineering, Vol. 40, pp. 2775-2790, 1997

9. A. A. Shabana, H. A. Hussien, and J. L. Escalona, "Application of the Absolute Nodal

Coordinate Formulation to Large Rotation and Large Deformation Problems", Journal

of Mechanical Design, Vol. 120, pp. 188-195, 1998

10. ADAMS Reference Manual, Mechanical Dynamics, 2301 Commonwealth Blvd, Ann

Arbor, MI 48105.

11. A. A. Shabana, Dynamics of Multibody Systems, John & Wiley, New York, 1989.



12. R. A. Wehage and E. J. Haug, "Generalized Coordinate Partitioning for Dimension

Reduction in Analysis of Constrained Dynamic Systems", Journal of Mechanical Design,

Vol. 104, pp. 247-255, 1982.

13. P. E. Nicravesh, Computer-Aided Analysis of Mechanical systems, Prentice-Hall, 1988

14. I. S. Duff, A. M. Erisman, and R. K. Reid, Direct Methods for Sparse Matrices,

Clarendon Press, Oxford, 1986

15. ANSYS Reference Manual, ANSYS, Inc., Southpointe 275 Technology Drive,

Canonsburg, PA 15317.

3-25



4

GENERALIZED RECURSIVE FORMULATION FOR FLEXIBLE MULTIBODY DYNAMICS

4.1. INTRODUCTION

The equations of motion for the general constrained mechanical systems were derived in terms of the relative coordinates by Wittenburg [1]. The velocity transformation method with the graph theory was employed to transform the equations of motion in the Cartesian coordinate space to the joint space systematically. Hooker [2] proposed a recursive formulation for the dynamic analysis of a satellite which has a tree topology. It was shown that the computational complexity of the formulation increases only linearly to the number of bodies. Fetherstone [3] used the recursive formulation to perform the inverse dynamic analysis of manipulators. Bae and Haug [4] further developed the formulation for constrained mechanical systems by using the variational vector calculus. The recursive formulation was applied to linearize the equations of motion [5]. Recursive formula for each term in the equations of motion was directly derived, using the state vector notation. Similar approach was taken in Ref. 6 to implement the implicit BDF integration with the relative coordinates. Since the recursive formulas were derived term by term, the resulting equations and algorithm became much complicated. To avoid the complication, the equations of motion were derived in a compact matrix form by using the velocity transformation method in Ref [7]. The generalized recursive formula for each category of the computational operations was developed and applied whenever such a category was encountered. This research applies the generalized recursive formulas for the multibody flexible dynamics.

Shabana [8] presented a coordinate reduction method for multibody systems with flexible components. The local deformation of a flexible component was expressed in terms of the nodal coordinates and was then spanned by a set of mode shapes obtained from a mode analysis. A fully Cartesian coordinate



formulation for rigid multibody dynamics by Jalon [9] was extended to the flexible body dynamics by Vukasovic and Celigueta [10]. Nonlinearity associated with an orientational transformation matrix was relieved by defining all necessary vectors for the equations of motion and constraints as the generalized coordinates. Variational equations of motion for flexible multibody systems were derived in Ref. 11. The variational approach was applied to extend the rigid body recursive formulation to flexible multibody systems. An extended kinematic graph concept was employed to develop a new recursive formulation for the dynamic analysis of flexible multibody systems by Lai and Haug [12]. Cardona and Geradin [13] dealt with substructuring for dynamic analysis of flexible multibody systems. The joint coordinates and the finite element method were employed for the flexible body dynamics by Nikravesh [14]. Pereira [15] presented a systematic method for deriving the minimum number of equations of motion for spatial flexible multibody systems.

Contrast to implementation of a rigid body dynamic analysis program, it is generally complicated to implement a flexible body dynamic formulation and to expand it for a general purpose program, regardless of whatever formulation has been chosen. This is because the flexible body dynamic formulations handle additional generalized coordinates to these of the rigid body dynamics. One of the most tedious works involved with the implementation of the flexible body dynamics is to build a set of joint and force modules. Whenever a new force or joint module is developed for the rigid body dynamics, the corresponding module for the flexible body dynamics has to be formulated and programmed again. In order to avoid such a repetitive process, this investigation proposes a concept of virtual body and joint. The relative coordinate kinematics and the virtual body concept are presented in section 2. A graph representation of flexible multibody systems is presented in section 3. The forward recursive formula and backward recursive formula respectively are treated in sections 4 and 5. A solution method of the equations of motion for a flexible body system is presented in section 6. Flexible slider crank mechanism is dynamically analyzed by using the proposed method to show its validity in section 7. Conclusions are drawn in section 8.

4-3

4.2. RELATIVE COORDINATE KINEMATICS OF TWO CONTIGUOUS

FLEXIBLE BODIES

4.2.1 COORDINATE SYSTEMS AND VIRTUAL BODIES

Figure 1 Two adjacent flexible bodies

The ZYX −−

O

frame is the inertial reference frame and the frame is the body reference frame in Fig. 1. Velocities and virtual displacements of point in the

zyx ′−′−′

ZYX −− frame are respectively defined as

ωr

(1)

and

ωrδδ

(2)

Their corresponding quantities in the zyx ′−′−′ frame are respectively defined as

=

′′

=ωArA

ωr

YT

T && (3)



and

=

′′

=πArA

πr

Zδ

δδδ

δT

T

(4)

where is the orientation matrix of the A zyx ′−′−′ frame with respect to the

ZYX −− frame. Two flexible bodies connected by a joint and their reference frames are shown in Fig. 1.

Suppose there exists a joint between the 111 iii zyx ′−′−′ and

frames, and a force applied at the origin of the 111 jjj zyx ′−′−′

222 jjj zyx ′−′−′ frame. Kinematic

admissibility conditions among the reference frames can be divided into two categories. One is the admissibility conditions between the two joint frames and the other is the admissibility conditions among the frames within a flexible body. These two types of conditions have been mixed in formulating the kinematic joint constraints and generalized forces in the previous works. As a result, every joint and force modules in a flexible multibody code, such as ADAMS [16] and DAMS [17], have been developed separately for rigid and flexible bodies. This would take long time for computer implementation and prone to coding errors. Especially, flexible body programming requires much more effort than rigid body programming does due to complexity associated with flexibility generalized coordinates and the strain energy.

Figure 2 Two adjacent flexible bodies and three virtual bodies

4-5

In order to minimize the programming effort, a concept of the virtual body is

introduced in this section. At every joint and force reference frames, a virtual rigid body, whose mass and moment of inertia are zero, is introduced. The virtual body and the original flexible body are then connected by a virtual joint. As an example, three virtual rigid bodies are introduced for two adjacent deformable bodies as shown in Fig. 2. Note that the flexible bodies have no joint or applied force except the virtual joints which are represented by the kinematic admissibility conditions among the flexible body frame and the virtual body frames. Therefore, the joint and force modules are developed only for rigid bodies and one flexible body joint of the virtual joints to be added in the joint module. The recursive kinematic relationships representing the admissibility conditions of the flexible body joint are formulated in the following subsections.

4.2.2 RELATIVE KINEMATICS FOR A FLEXIBLE BODY JOINT

Figure 3 Flexible body joint between a flexible body and a virtual body

A virtual body is always connected to the original flexible body by a flexible body joint. Origin of the virtual body reference frame in Fig. 4 can be expressed as follows:



)( )1()1(011 iiiiiii −−−− ′+′+= usArr (5)

where and ii )1(0 −′s ii )1( −′u are the undeformed location vector and deformation

vector of the origin of the virtual body with respect to the flexible body reference frame, is the orientation matrix of the flexible body reference frame. The deformation vector

1−iA

ii )1( −′u can be spanned by linear combination of a set of

mode shapes [8] as f

iiRiii )1(1)1( −−− =′ qΦu (6)

where is a modal matrix whose columns consist of the translational mode

shapes and superscript f in denotes the modal coordinate vector. Subscripts

and denote the generalized coordinate between the and i body reference frames.

Ri 1−Φ

1−i

fii )1( −q

i i 1−

The angular velocity in the local reference frame is obtained as follows

fiii

Tiii

Tiii )1(1)1(1)1( −−−−− +′=′ qΦAωAω &θ (7)

where is used. Differentiating Equation (5) and multiplying by

yields i

Ti

Tii AAA )1()1( −− =

TiA

f

iiiT

iiiiiT

iiiT

iii )1(1)1(1)1()1(1)1(~

−−−−−−−− +′′−′=′ qΦAωsArAr &&& θ (8)

where iiiiii )1()1(0)1(~

−−− ′+′=′ uss , symbol with tilde denotes skew symmetric matrix

which consists of their vector elements, and wide tilde ω are used. Combining Equations (7) and (8) yields the following recursive velocity equation for a flexible body joint.

iii ωAA ′= ~&i′

f

iif

iiif

iii )1(2)1(11)1( −−−− += qBYBY & (9) where

4-7

=

′−=

−−

−−−

−

−−−−

θ1)1(

1)1(2)1(

)1(

)1()1()1(1)1(

~

iT

ii

Ri

Tiif

ii

Tii

iiT

iiT

iifii

ΦA

ΦA

A0sAA

B

B

(10)

It is important to note that matrices and are function of only

modal coordinates of the flexible body i-1. As a result, further differentiation of the matrices and B in Equation (9) with respect to other than bf

yields null. This property will play a key role in simplifying recursive

formulas in sections 4 and 5.

fii 1)1( −B f

ii 2)1( −B

fii 1)1( −B f

ii 2)1( −

fii )1( −q

Equation (9) defines the kinematic relationships between an inboard flexible body and an outboard rigid body. The kinematic relationships between an inboard rigid body and an outboard flexible body can be derived similarly. Similarly, the recursive virtual displacement relationship between a flexible body and a virtual body is obtained as follows

f

iif

iiif

iii )1(2)1(11)1( −−−− += qBYBY & (11) where

−

−′−=

′=

−

−−−−

−

−−−−

θ

θ

1

11)1(2)1(

)1(

)1()1()1(1)1(

~

~

i

Riiiir

ii

Tii

Tiiii

Tiir

ii

Φ

ΦΦsB

A0AsA

B

(12)

4.2.3 RELATIVE KINEMATICS FOR A RIGID BODY JOINT

The recursive velocity relationship for a rigid body joint connecting two rigid bodies can be derived by following the similar steps as in Equations (5)-(9) as

r

iir

iiir

iii )1(2)1(11)1( −−−− += qBYBY & (13)



where superscript r denotes the generalized co-ordinate from a rigid body joint and

′′′+′

=

′−′+′−=

−−

−−−−−−−

−

−−−−−−−−

−

iiT

ii

iiT

iiiiiiiiT

iirii

Tii

Tiiiiiiiiii

Tii

Tiir

ii

ii

)1()1(

)1()1()1()1()1()1(2)1(

)1(

)1()1()1()1()1()1()1(1)1(

)~)~((

)~~~(

)1(

HAHAsAdA

B

A0AsAdsAA

B

q

(14)

where is determined by the axis of rotation. Node that the matrices are

function of only . ii )1( −′H B

rii )1( −q

4.2.4 GRAPH REPRESENTATIONS OF MECHANICAL SYSTEMS

The graph theory was used to automatically preprocess mechanical systems

having various topological structures in References [1, 4]. A node and an edge in a graph have represented a body and a joint, respectively. The preprocessing, based on the graph theory, yields the path and distance matrices that are provided to automatically decide computational sequences. Two computational sequences are required in a general purpose program. One is the forward path sequence starting from the base body and moving towards the terminal bodies. The other is the backward path sequence starting from the terminal bodies and moving towards the base body.

Figure 4 Flexible slider crank mechanism

4-9

Figure 5 Graph representation and computational sequence

In order to derive systematically the recursive formulas, bodies in a graph are divided into four disjoint sets (associated with a generalized coordinate ) as follows :

kq

)( kqI =adjacent outboard body of the joint having as its generalized co-ordinate

kq

)( kqII =all outboard bodies of , excluding all bodies in )( kqI )( kqI

)( kqIII =all bodies between the base body and the inboard body of , including the base and inboard bodies and excluding all bodies in

)( kqI

)( kqI

)( kqIV = the complementary set of )()()( kkk qqq IIIIII ∪∪



As an example, the graph theoretic representation and computational path

sequences of the system in Fig. 4 are shown in Fig. 5. The four disjoint sets for the system in Fig. 5, if belongs to the joint between bodies 3 and 4, are

=body 4, =bodies 5, 6, and 7, =bodies 1, 2, and 3, =bodies 8, 9, 10, 11, and 12

kq

)( 34qI

( 34qIV

)( 34qII )( 34qIII

)

4.3 FORWARD RECURSIVE FORMULAS

4.3.1 GENERALIZATION OF THE VELOCITY RECURSIVE FORMULA

Generalization of the velocity recursive formula can be achieved by computational equivalence between the recursive method and the velocity transformation method. The velocity Y for all bodies in a system can be obtained by repetitive symbolic substitutions of the recursive formula in Equations (9), (11) and (13), depending on the type of a joint, along the forward path sequence of a graph and by appending the trivial equation of as follows :

ff qq && =

qBqq

I0BB

qY

Y &&

&

&≡

=

= f

rzfzr

f (15)

where and q are the relative and modal co-ordinates vectors for a system,

respectively. The dimension of , and are, respectively, assumed to

be ,

rq

n

f

Y rq& fq&

nc r , and . The velocity nf nfncR +∈Y with a given can be evaluated either by using Equation (15) obtained from symbolic substitutions or by using (9), (11) and (13) with recursive numeric substitution of 's. Since both formulas give an identical result and recursive numeric substitution is proven to be more efficient [4], matrix multiplication with a given will be actually evaluated by using Equations (9), (11) and (13). Since q in

nfnrR +∈q&

iY

qB & q&

&

4-11

Equation (15) is an arbitrary vector in nfnrR + , Equations (9), (11), (13) and (15) which are computationally equivalent, are actually valid for any vector such that

BxxX

=

BXBX +

q)(Bx

i )1( −+ B

i

ii ( 11 −X

nfnrR +∈x

X

≡ (16)

and

X = (17)

where nfncR +∈X is the resulting vector of multiplication of and and B matrices depend on a joint type. As a result, transformation of

B xnfncR +∈Y into

nfncR +∈Bx is actually calculated by recursively applying Equation (17) to achieve computational efficiency in this research.

4.3.2 RECURSIVE FORMULA FOR qX =

Equation (17) is partially differentiated with respect to for

to obtain the recursive formula for as follows. kq

)(,...,1 nfnrk += q)(Bx

iiqiiqiiiqiiqi kkkk )1(2)1(1111)1( )()()()( −−−−− += XBXXBX (18)

Since matrices depend only on the generalized coordinates for joint , their partial derivatives with respect to generalized coordinates other than

become null. In other words, the partial derivatives become null if

does not belong to set . If body is an element of set , Equation (18) becomes

Bii )1( −

ii )1( −q kq

)( kqI )( kqII

(19)

kk qiqi ))( )1( −= BX



Figure 6 Computation sequence of

iiqY If body belongs to set , is not affected by . As a

result, Equation (18) is further simplified as follows. i )()( kk qq IVIII ∪ iX kq

0X =

kqi )( (20)

4-13

There are two recursive formulas in the case of body . If body is an element of set , body

)( kqI∈i i

)( kqI 1−i is naturally its inboard body and belongs to set . Equation (18) becomes )( kqIII

iiqiiiqiiqi kkk )1(2)1(11)1( )()()( −−−− += XBXBX (21)

If bodies and are elements of set , the recursive formula in Equation (18) is expressed as follows:

1−i i )( kqI

iiqiiqiiiiqiiqi kkkk )1(2)1()1(1)1(11)1( )()()()( −−−−−− ++= XBXBXBX (22)

As an example, the recursive formula in Equation (19)-(22) can be applied to compute

34qY for the system in Fig. 4, as shown in Fig. 6. Note that since the

recursive formulas for and can be obtained similarly, they are

omitted.

Bx q)(Bx

4.4 BACKWARD RECURSIVE FORMULAS

4.4.1 GENERALIZATION OF THE FORCE RECURSIVE FORMULA

A generalized recursive formula for transformation of nfnrR +∈x into a new vector BxX = in nfncR +

G is derived in section 4. Inversely, it is often necessary

to transform a vector in nfncR + into a new vector in GBTg = nfnrR + . Such a transformation can be found in the generalized force computation in the joint space with a known force in the Cartesian space. The virtual work done by

is obtained as follows. nfnc+R∈Q

[ ]

≡= f

cfTTT

QQ

qZQZW δδδδ (23)

where Zδ must be kinematically admissible for all joints in a system and cQ



and are the Cartesian and modal forces, respectively. Substitution of virtual displacement relationship into Equation (23) yield

fQ

Q

**)( ffTrrTfcT

zrfTcT

zrrT QqQqQQBqQBqW δδδδδ +=++= (24)

where and Q . Equation (24) can be written in a

summation form as

cTzr

r QB≡* fcTzf

f QQB +≡*

∑∑∈+

++∈+

++ +=fjtsii

fii

fTii

rjtsii

rii

rTii

)1(

*)1()1(

)1(

*)1()1( QqQqW δδδ (25)

where rjts and fjts respectively denote all rigid body joints and all flexible body joints.

On the other hand, the symbolic substitution of the recursive virtual displacement relationship into Equation (23) along the chain (starting from the terminal bodies toward inboard bodies) and the reorganization of the equation about the virtual relative displacement and modal displacement yield

( )

( )∑ ∑

∑ ∑

∈+ ∈+++++

∈+ ∈++++

++

+=

+

+

fjtsii qli

ci

fTii

fii

fTii

rjtsii qli

ci

rTii

rTii

ii

ii

)1( )(112)1()1()1(

)1( )(112)1()1(

)1(

)1(

I

I

SQBQq

SQBqW

δ

δδ

(26)

where

∑++

++∈

+++

+

+≡

+≡

)(221

1

)2)(1(

)2)(1()(

body terminala is 1 if

ii

iiql

ici

Ti

i i

I

SQBS

0S (27)

The recursive formula for bf and is obtained by equating Equations (25) and (26) as follows:

*fQ *rQ

∑+∈

+++++ ++=)(

112)1()1(*

)1()1( iiql

ici

Tiiiiii

I

)S(QBQQ (28)

4-15

where for a rigid body joint and for a flexible body joint connecting

an inboard flexible body and an outboard virtual body, and for a

flexible body joint connecting an inboard virtual body and an outboard flexible body, and is defined in Equation (27).

0=+ )1(ii

1+iS

Qf

iiii )1()1( ++ = QQ

Since Q in Equation (23) is an arbitrary vector in nfncR + , Equations (23) and (28)are valid for any vector in G nfncR + . As a result, the matrix multiplication of is actually evaluated to achieve computational efficiency in this research by

GBT

∑+∈

+++++ ++=)(

112)1()1()1()1( iiqIl

ici

Tiiiiii )S(GBGg (29)

where is the result of and is defined as in Equation

(28) and

g GBT)1( +iiG )1( +iiQ

body terminala is 1 if1 +≡+ ii 0S (30)

∑++∈

+++++ +≡)(

221)2)(1(1)2)(1(

)(iiql

ici

Tiii

I

SQBS (31)

Recursive formula in Equation (29) must be applied for all joints in the

backward path sequence to obtain where is a constant vector in GBg T= GnfncR + .

4.4.2 RECURSIVE FORMULA FOR

kk qT

q )( GBg =

The Recursive formula for is obtained by replacing by in

Equation (29) and kq

T )( GB i 1−i

1+i by 1−i in Equation (31) and taking partial derivative with respect to yield kq

∑

∑

−

−

∈−

∈−−−

++

++=

)(2)1(

)(2)1()1()1(

)1(

)1(

)()()(

ii

k

ii

kkk

qlqi

ci

Tii

qli

ciq

Tiiqiiqii

I

I

)S(GB

)S(GBGg

(32)



∑∑−− ∈

−∈

−− +++=)(

1)1()(

1)1()1()1()1(

)()(ii

k

ii

kkql

qici

Tii

qli

ciq

Tiiqi

II

)S(GB)S(GBS (33)

Since is a constant vector, . If

, matrices are not functions of . Therefore, their partial derivatives with respect to become null. As a result, Equations (32) and (33) can be simplified as follows.

nfncR +∈G

)( kq IVIII ∪

0G =kq

kq)()( kk qqi II ∪∈ B

kq

∑−∈

−− =)(

2)1()1()1(

)()(ii

kkql

qiT

iiqiiI

SBg (34)

∑−∈

−− =)(

)1(1)1(

)()(ii

kkqIl

qiT

iiqi SBS (35)

Since for the terminal bodies, for .

Thus, for , Equation (34) becomes

0S =kqi )(

)( kqi II∈

0S =kqi )( )()( kk qqi IVII ∪∈

)( kqIV∪

0g =− kqii )( )1( (36)

There are two recursive formulas in the case of body . If body

and body

)( kqi I∈

)( kqi I∈ 1+i belongs to set , and . Thus, Equation

(32) and (33) become

)( kqII 0S =kqi )(

∑−∈

−− +=)(

2)1()1()1(

)()(ii

kkql

iciq

Tiiqii

I

)S(GBg (37)

∑−∈

−− +=)(

1)1()1()1(

)()(ii

kkql

iciq

Tiiqi

I

)S(GBS (38)

where must be saved when is computed. This recursive formula can

be applied to compute . As an example, iS GBT

qT )( GB ( )

3434)( q

Tq GBg =

34qqk =

for the system

in Fig. 4 is obtained, as shown in Fig. 7 for the case of . Note that the components of ( )

34qg are either zero or simple to compute.

4-17

Figure 7 Computation sequence of .

iiii qq )()( BGg =

4.5. THE GOVERNING EQUATIONS OF SOLUTION

4.5.1 IMPLICIT INTEGRATION OF THE EQUATIONS OF MOTION

The dynamic equations of motion for a constrained mechanical system in the joint space have been obtained in Reference [1] by the velocity transformation



method as follows.

)QλΦY(MBF Z −+= TT & (41)

where and , respectively, denote the cut joint constraint and the

corresponding Lagrange multiplier. The vector including external forces, strain energy terms, and velocity induced forces.

Φ λ

M is a mass matrix and is a force

The equations of motion, the constraint equations, vq =& , and constitute the differential algebraic equations(DAE). Application of 'tangent space method' in Reference [19] to the DAE yields the following nonlinear system of equations

av =&

0

),z,v,(qΦ)v,(qΦ

),Φ(q),λ,a,v,F(q)βa(vU)βv(qU

)H(p =

++++

=

nnnn

nnn

nn

nnnnn

nnT

nnT

n

tt

tt

&&

& ,

200

100

ββ

(42)

where , , , and are determined by the coefficients of the BDF. The must be chosen such that the augmented square matrix

is nonsingular. Applying Newton's method to solve the nonlinear system

in Equation (42) yields

[ TTn

Tn

Tn

Tn

Tn λavqp ,,,=

0U

] 0β 1β 2β

qΦUT

0

Hp)H(p −=∆n (43)

,...3,2,1,)()1( =∆+=+ iin

in ppp (44)

where

4-19

=

0ΦΦΦ00ΦΦ000Φ

FFFF0UU000UU

)H(p

avq

vq

q

λavq

&&&&&&

&&

TT

TT

n

000

000

ββ

(45)

Since and are highly nonlinear functions of q, and qF qΦ av, λ

, some cautions must be taken in deriving the non-zero expressions in matrix so

that they can be efficiently evaluated. pH

4.5.2 APPLICATION OF THE GENERALIZED RECURSIVE FORMULAS

A set of the generalized recursive formulas has been developed in the sections

3 and 4. This section shows how these formulas can be utilized to efficiently compute the in H in Equation (45). Inspection of reveals that partial

derivatives of F , , , , and are needed to be computed.

Only the is presented in this section and the rest can be derived similarly.

qF p

vF

pH

q aF qΦ qΦ& qΦ&&

qF

In Equation (41), differentiation of matrix with respect to vector q results in a three dimensional matrix. To avoid the notational complexity for the three dimensional matrix, Equation (41) is differentiated with respect to each generalized coordinate one by one. Thus,

B

kq

nfnrkkk

kk

qT

qT

TTqq

+=−++

−+=

,...,3,2,1),()( )QλΦY(MB

)QλΦY(MBF

Z

Z

&

&

(46)

Since the term )QλΦZ −T( can be easily expressed in terms of the Cartesian

coordinates, kq

T )QλΦZ −( is obtained by applying the chain rule as follows.


nfnrkkT

qT

k+=−=− ,...,3,2,1,(( B)QλΦ)QλΦ ZZZ (47)


where Bq/Z =∂∂

GBTqk

is used and B denotes the th column of the matrix . The first term in Equation (46) can be obtained by applying the recursive formula for with

k k B

)QλΦY(MG Z −+= T&

. Collection of for allk constitutes )QλΦZ −T( , which is equivalent to )QλΦZ −T( . Matrix T

Z)

G

TZ QλΦ −(

consists of nc+nf column vectors in . Therefore, the application of ,

where is each column of matrix

nfncR + BTqk

G TZ

TZ )QλΦ −( , yields the numerical result of

)QλΦZ −T( . Finally, the second term in Equation(46) is also obtained by applying

, where GBTqk

)QλΦY(MG Z −+= T& and kqY& is recursively obtained.


Dynamic analysis of a flexible slider crank mechanism is presented in order to validate the results from the proposed method. The example problem is solved by using both the proposed method and the nonlinear approach developed by Simo [19].

The system consists of two rigid bodies and one flexible body, as shown in Fig. 4. Length, cross-sectional area, and area moment of inertia of the elastic crank are 0.4 m, 0.0018 m2, and 1.35×10-7 m4, respectively. The crank is modeled by using 10 two-dimensional elastic beam elements of equal length. The material mass density of the beam is 5540.0 kg/m3 and its Young's modulus is 1.0 times 109 N/m2. Vibration analysis of the crank is carried out with fixed-free boundary condition and the resulting mode shapes are shown in Fig. 8, 9. Four mode shapes are selected to span the deformation of the crank. As a result, the system has 5 degrees of freedom.

Dynamic analysis is performed for 5 sec under the constant acceleration condition of the joint between the ground and the body 1. The acceleration, displacement, and relative deformation of the pin joint connecting the crank and the coupler both from the proposed method and the nonlinear approach[19] are shown in Figs.10,11 and 12, respectively. Note that since the results from both models are almost identical as shown in these figures, the proposed implementation methods using rigid virtual body can be validated.

4-21

Figure 8 Mode shapes of the crank

Figure 9 Mode shapes of coupler



Figure 10 acceleration of Y 1P

Figure 11 Relative deformation of 1P

4-23

Figure 12 The strain energy of crank

4.7. CONCLUSION

This research extends the generalized recursive formulas for the rigid

multibody dynamics to the flexible body dynamics using the backward difference formula(BDF) and the relative generalized coordinate. When a new force or joint module is added to the general purpose program in the relative coordinate formulations, the modules for the rigid bodies are not reusable for the flexible bodies. In order to relieve the implementation burden, a virtual rigid body is introduced at every joint and force reference frames and a virtual flexible body joint is introduced between two body reference frames of the virtual and original bodies. The notationally compact velocity transformation method is used to derive the equations of motion in the joint space. The terms in the equations of motion which are related to the transformation matrix are classified into several categories each of which recursive formula is developed. Whenever one category is encountered, the corresponding recursive formula is invoked. Since computation time in a relative coordinate formulation is approximately proportional to the number of the relative coordinates, computational overhead due to the additional virtual bodies and joints is minor. Meanwhile, implementation convenience is dramatically improved.



REFERENCE

1. J. Wittenburg, Dynamics of Systems of Rigid Bodies, B. G. Teubner, Stuttgart, 1977.

2.W. Hooker, and G. Margulies, "The Dynamical Attitude Equtation for an n-body Satellite",

it Journal of the Astrnautical Science, Vol. 12, pp. 123-128, 1965.

3. R. Featherstone, "The Calculation of Robot Dynamics Using Articulated-Body

Inertias",it Int. J. Roboics Res., Vol 2, pp. 13-30, 1983.

4. D. S. Bae and E. J. Haug, "A Recursive Formulation for Constrained Mechanical

System Dynamics: Part II. Closed Loop Systems", it Mech. Struct. and Machines, Vol. 15,

No. 4, pp. 481-506.

5. T. C. Lin, and K. H. Yae, "Recursive Linearization of Multibody Dynamics and

Application to Control Design", it Technical Report R-75, Center for Simulation and


Mathematics, The University of Iowa, Iowa City, Iowa, 1990.

6. Ming-Gong Lee and E. J. Haug, "Stability and Convergence for Difference

Approximations of Differential-Algebraic Equations of Mechanical System Dynamics", it

Technical Report R-157, Center for Simulation and Design Optimization, Department of

Mechanical Engineering, and Department of Mathematics, The University of Iowa, Iowa

City, Iowa, 1992.

7. D. S. Bae, J. M. Han, H. H. Yoo, and E. J. Haug. "A Generalized Recursive Formulation

for Constrained Mechanical Systems", it Mech. Struct. and Machines, To appear.

8. A. A. Shabana, "Substructure Synthesis Methods for Dynamic Analysis of Multibody

Systems", it Computers & Structures, Vol. 20. No. 4, pp 737-744, 1985.

9. J. Garcia de Jalon, J. Unda, and A. Avello, "Natural Coordinates for the Computer

Analysis of Three-Dimensional Multibody Systems", it Computer Methods in Applied


10. N. Vukasovic, J. T. Celigueta, J. Garcia de Jalon, and E. Bayo, "Flexible Multibody

Dynamics Based on a Fully cartesian System of Support Coordinates", it Journal of

Mechanical Design, Vol. 115, pp. 294-299, 1993.

11. S. S. Kim and E. J. Haug, "A Recursive Formulation for Flexible Multibody

4-25

dynamics:Part I, Open loop systems", it Comp. Methods Appl. Mech.Eng, Vol. 71,

pp.293-314, 1988.

12. H. J. Lai, E. J. Haug, S. S. Kim, and D. S. Bae. "A Decoupled Flexible-Relative

Coordinate Recursive Approach for Flexible Multibody Dynamics", it International

Journal for Numerical Methods in Engineering, Vol. 32, pp.1669-1689, 1991.

13. A. Cardona and M. Geradin, "Modelling of Superelements in Mechanism Analysis", it

International Journal for Numerical Methods in Engineering, Vol. 32, pp.1565-1593,

1991.

14. P. E. Nikravesh and A. C. Ambrosio, "Systematic Construction of Equations of Motion

for Rigid-Flexible Multibody Systems Containing Open and Closed Kinematic Loops", it

International Journal for Numerical Methods in Engineering, Vol. 32, pp.1749-1766,

1991.

15. M. S. Pereira and P. L. Proenca, "Dynamic Analysis of Spatial Flexible Multibody

Systems Using Joint Coordinates", it International Journal for Numerical Methods in

Engineering, Vol. 32, pp.1799-1812, 1991.

16. ADAMS Reference Manual, Mechanical Dynamics, 2301 Commonwealth Blvd, Ann

Arbor, MI 48105.

17. A. A. Shabana, Dynamics of Multibody Systems, John & Wiley, New York, 1989.

18. Jeng Yen, E. J. Haug, and F. A. Potra, "Numerical Method for Constrained Equations

of Motion in Mechanical Systems Dynamics", it Technical Report R-92, Center for

Simulation and Design Optimization, Department of Mechanical Engineering, and

Department of Mathematics, The University of Iowa, Iowa City, Iowa2 1990.

19. J. C. Simo, and L. Vu-Quoc, "On the Dynamics of Flexible Beams Under Large Overall

Motions-The Plane Case: Part I", it Journal of Applied Mechanics, Vol. 53, pp. 849-854,

1986.



5

RELATIVE NODAL METHOD FOR LARGE DEFORMATION PROBLEM

5.1. INTRODUCTION

Geometrically nonlinear analyses[1-4] have been investigated by many researchers. Their equations of equilibrium are based on either the total Lagrangian formulation or the updated Lagrangian formulation. Since all displacements are referred to the initial configuration in the total Lagrangian formulation, the resulting equations of equilibrium are relatively simple. However, if a structure undergoes a large displacement, some difficulties may be encountered due to the nonlinearity associated with rotation. All displacements are referred to the last calculated configuration in the updated Lagrangian formulation and the rotational nonlinearity is relieved if the load increment is small. The same difficulties as the total Lagrangian formulation can be encountered in the case of a large load increment.

Avello[5] referred kinematic variables relative to the initial configuration and he expressed the strains in a moving frame. Therefore, the strains were invariant for finite rigid body deformations. Shabana[6-8] presented an absolute nodal coordinate formulation for flexible multibody dynamics. All finite elements were reformulated. Shimizu[9] considered the rotary inertia effects. This method is based on the absolute nodal coordinate formulation.

Moving reference frame approaches were proposed by some researchers in Refs. 10-14. A moving reference frame is introduced to represent a finite rigid body motion. Deformation at a point of a flexible body was super-imposed on the rigid body motion.



5.2. RELATIVE DEFORMATION KINEMATICS

5.2.1 GRAPH THEORETIC REPRESENTATION OF A STRUCTURE

This paper proposes to use the relative nodal displacements in formulating the equations of equilibrium. Since the absolute nodal deformations are obtained by accumulating the relative deformations along a path, element connectivity information must be identified prior to generating the equations of equilibrium for a general system. Therefore, a topology analysis must be carried out for a structural system discretized into many finite elements.

0 1 2 3 4

Figure 1(a) A cantilever beam with five nodes

Forward path sequence

Backward path sequence

0 1 2 3 4

Figure 1(b) Graphic theoretic representation for the cantilever beam

The discretized systems can be represented by a graph. A node and an element

are represented by a node and an edge in the corresponding graph, respectively. As an example, the graph theoretic representation for the system in Fig. 1(a) is shown in Fig. 1(b). If a structure possesses a loop in its graph theoretic representation, it is called as a closed loop system. Otherwise, it is called as an open loop system.

A spanning tree denotes a graph which does not have a closed loop. A node which does not have a child node is called as a terminal node. A node which does not have a parent node is called as a base node. The terminal node and the base node for the system in Fig. 1(b) are nodes 4 and 0, respectively. Two

5-3

computational sequences must be defined in the proposed relative displacement formulation. One is the forward path sequence which traverses a graph from the base node towards the terminal nodes. The other is the backward path sequence which is the reverse of the forward path sequence. Two sequences for the graph in Fig. 1(a) are shown in Fig. 1(b).

5.2.2 KINEMATIC DEFINITIONS

Consider a system consisting of two beam finite elements as shown in Fig.

2(a) and (b). Nodes i and are assumed to be inboard nodes of nodes and in a graph, as shown in Fig. 2(b), respectively. is the inertial reference frame and

1− i

−

i1+i ZYX −−

kkk zyx − ),( jik = is the nodal reference frame

attached to a node , and is a position vector of the node k kr k . is the reference frame attached to a node and the first

subscript denotes the inboard node number of the second subscript . The orientation of coincides with that of in

the undeformed state. The absolute nodal displacements measured in the frame have been solved for in the conventional finite element analysis

methods(see Refs. 1-4). In contrast to conventional methods, the relative nodal displacements measured in its inboard nodal reference frame are solved in this paper.

iiiii )1()1)1( zx −−− −

1−i

i 1(x −

X −

i(y−

ZY −

i

)1()1 zy − −− i

i

( −ii)i 1(z −ii )1(y −i) −− )1(x −i

ix

iz

iy

1x −i

1z −i

1y −i

i-1

i

X

Z

Y

ir1−ir

ii )1(x −

ii )1(z −

ii )1(y −

i+1


Figure 2(a) Two finite beam elements


Forward path sequence

Backward path sequence

LL ii-1 i+1

Figure 2(b) Graphic theoretic representation for the beam elements

The generalized coordinates for the relative nodal position and orientation

displacements of a node are denoted by u and , respectively. The

nodal position and orientation of node in the

')1( ii−

i

')1( ii−Θ

YX Z−− frame can be expressed in terms of these of node 1−i and the relative nodal displacements as follows:

( )')1(

'0)1()1()1( iiiiiii −−−− ++= usArr (1)

iiiiiiii )1('

)1()1(1 )( −−−− Θ= CDAA (2) where

[ Tiiiiiiii

'3)1(

'2)1(

'1)1(

')1( −−−− =Θ θθθ ] (3)

In Eqs.(1) and (2), kA ),1( iik −= denotes the transformation matrix for nodal reference frame , denotes the constant transformation matrix from

to , denotes the location vector of node

measured in in the undeformed state, and denotes the

deformation vector of node relative to the nodal frame

k

i)1−

) −

ii )1( −

iii )1()1 z −− −

)1(z −− i

i

C

i(y−

)1(y −i

iii zyx −− i(x

1( −i

'0)1( ii−s i

x ')1( ii−u

1−i

ii )1(x − −

. is the

transformation matrix due to a rotational displacement of

relative to the nodal frame

ii )1( −D

ii )1(y − i)i 1(z −−

1−i and can be expressed by the 1-2-3 Euler angle as

)()()( '3)1(3

'2)1(2

'1)1(1)1( iiiiiiii −−−− = θθθ DDDD (4)

Taking a variation of Eq. (1) yields

( ) ')1()1(

')1(

')1(

'0)1()1(

')1()1(

' ~~ii

Tiiiiiii

Tiii

Tiii −−−−−−−− ++−= uAusArAr δδδδ π (5)

5-5

where a symbol with tilde denotes a skew symmetric matrix which consists of its vector elements, and is defined as ii )1( −A

iTiii AAA )1()1( −− = (6)

The virtual rotation relationship between nodes and i 1−i is given as

')1()1()1(

')1()1(

'iiii

Tiii

Tiii −−−−− Θ+= δδδ HAA ππ (7)

where

−=

−−−

−−−

−

−

)cos()cos()sin(0)cos()sin()cos(0

)sin(01

'2)1(

'1)1(

'1)1(

'2)1(

'1)1(

'1)1(

'2)1(

)1(

iiiiii

iiiiii

ii

ii

θθθθθθ

θH (8)

Combining Eqs.(5) and (7) yields the following recursive virtual displacement equation for a pair of contiguous elements.:

iiiiiiii )1(2)1()1(1)1( −−−− += qBZBZ δδδ (9)

where

[ ] ),1(,' iikTTk

Tkk −== πδδδ rZ (10)

[ ]TTii

Tiiii

')1(

')1()1( −−− = Θδδδ uq (11)

+−

= −−

−

−− I0

usIA0

0AB

)~~( ')1(

'0)1(

)1(

)1(1)1(

iiiiT

ii

Tii

ii (12)

=

−−

−−

iiT

ii

Tii

ii)1()1(

)1(2)1( H

IA0

0AB (13)

It is important to note that matrices and are only functions of the

relative displacement between nodes 1)1( ii−B 2)1( ii−B

ii )1( −q 1−i and i .

The virtual displacement relationship between the absolute and relative nodal coordinates for the whole system can be obtained by repetitive application of Eq. (9) along a chain in a graph. As an example, the virtual displacement relationship



for the Cartesian and relative coordinate systems in Fig. 1 is as follows.

qBZ δδ = (14) where

[ TTTTT4321 ZZZZZ δδδδδ = ]]

(15)

[ TTTTT34231201 qqqqq δδδδδ = (16)

=

342232341122231341012121231341

232122231012121231

122012121

012

BBBBBBBBBB0BBBBBB00BBB000B

B (17)

5.3. GOVERNING EQUATIONS OF EQUILIBRIUM

5.3.1 STRAIN ENERGY

The strain energy in a finite element having multiple nodes is affected only by the relative displacements of nodes relative to the inboard nodal frame of the element and is free from its rigid body motion. As a result, the variational form of the strain energy for a system can be obtained in a summation form as

∑=

−−− ==n

k

Tkkkk

TkkW

1)1()1()1( KqqqKq δδδ (18)

where qδ must be kinematically admissible for all constraints. Since the stiffness matrix is generated in the nodal reference frame, the strain energy due to a rigid body motion of a node does not appear in Eq. (18). The element stiffness matrix is contributed from linear and nonlinear terms as (see

Ref. 3) kk )1( −K

nL

kkL

kkkk )1()1()1( −−− += KKK (19) where

∫−

−−−− ΓΞΓ= kkl

kkL

kkT

kkL

kk dx)1(

0

*)1()1(

*)1()1(K (20)

5-7

∫+

−−−−− ΓΞΓ= )1(

0

*)1()1()1(

*)1()1( )(iil

kkkknL

kkT

kknL

kk dxqK (21)

In Eqs. (19) - (21), denotes a linear stiffness matrix, denotes a

nonlinear stiffness matrix, and l denotes the undeformed length of the

element between the nodes and . Note that the significance of

depends on the magnitude of . becomes negligible when the

magnitude of is small, which is true when the element size is small. It is

very difficult analytically to prove the significance of . As a consequence,

the significance of has been demonstrated through a numerical example

in § 5.

Lkk )1( −K

nLkk )1( −K

nLkk )1( −K

kk )1( −

kk )1( −q

1−k k nLkk )1( −K

nLkk )1( −K

kk )1( −qnL

kk )1( −K

5.3.2 EXTERNAL FORCE

The virtual work done by both nodal forces described in the absolute

nodal coordinate system and described in the relative nodal coordinate system is obtained as follows:

QR

RqQZ TTW δδδ += (22)

where Zδ must be admissible for the kinematic relationship between Zδ and

qδ . Substitution of qBZ δδ = into Eq. (22) yields

( ) *QqRQBq TTTW δδδ =+= (23) where

RQBQ += T* (24)



5.3.3 CONSTRAINT

cut0

23

4

5

1

Figure 3 A closed loop system

A nodal displacement is measured relative to its inboard nodal frame in the

proposed method. The relative nodal displacement can be defined only in structures having a tree topology. Therefore, if a structural system has a closed loop, it must be opened to form the tree topology. The cut joint method (see Ref. 12) is employed to treat the closed loops. A node in a closed loop is removed and the corresponding cut constraint equations are introduced to compensate for the removed node. As an example, Fig. 3 shows a closed loop system. The graphical representation of the system is presented in Fig. 4.

0

1 5

2 4

3

cut

Figure 4 Graphic representation of the system of the closed loop system in Fig. 3

5-9

0 2 31 4 5

Figure 5 Tree structure corresponding to the system in Fig. 3

A cut has been made at node 5 to form the tree structure shown in Fig. 5. The cut constraint can be formulated from the geometric compatibility relationships. From Eqs. (1) and (2), the position and orientation matrix of node 5 is obtained along the forward path sequence as

( ) *5

')1(

'0)1(

5

1)1(5 rusAr =+= −−

=−∑ kkkk

kk (25)

∏=

−− ==5

1

*5)1()1(05

kkkkk ACDAA (26)

where and are given by the boundary conditions at node 5. Since Eq. (26) comprises of nine dependent equations, only three are independent. The three independent constraint equations can be extracted by imposing perpendicularity between the axes of reference frames. As a result, the six independent constraint equations are given as

*5r *

5A

−

=

*5251

*5253

*5153

*55

aaaaaarr

Φ

T

T

T

(27)

where

[ ]5352515 aaaA = (28) [ ]*

53*52

*51

*5 aaaA = (29)

In Eqs. (28) and (29), and i5a *

5ia )3,2,1( =i denote the -th column vector of and , respectively.

i

5A *5A



5.3.4 EQUATIONS OF EQUILIBRIUM

For a closed loop system, relative deformation q is not independent, and q

must satisfy the constraint Eq. (27). Taking variation of Eq. (27) yields

0qΦΦ q == δδ (30)

The Lagrange multiplier theorem (see Refs. 12 and 15) can be applied to obtain the following equations of equilibrium for a constrained system:

( ) 0ΦQKqq q =+− λTT *δ (31)

where the qδ is arbitrary. Since qδ is arbitrary, its coefficient must be zero, which yields

( ) 0QΦKqqF q =−+= *, λλ T (32)

Since the number of equations is less than that of unknown variables in Eq. (32), the unknown variables cannot be determined. Thus, constraint equations given in Eq. (27) are supplemented to find the solution of and q λ . Deformations

can be obtained by solving Eqs. (27) and (32) simultaneously. Since the , ,

and in the equations are the nonlinear function of , can be solved by using Newton-Raphson method as

q

qΦ Φ*Q q q

−=

∆∆

ΦFq

0ΦΦF

q

qq

λ

T

(33)

where ( )

qqq QΦKF *−+= λT (34)

By solving Eq. (33), the improved solution of for the next iteration can be obtained as follows:

q

qqq ∆+= (35)

5-11

By using Eqs. (33) and (35), the iteration continues until the solution variance remains within a specified allowable error tolerance. Before solving Eq. (33), it is necessary to calculate . However, the calculation of F is numerically

difficult and tedious. qF q

In order to save computing time in solving Eq. (33), some numerical approximation techniques may be applied. As an example, the coefficient matrix of Eq. (33) may remain near constant if the variation of is small, which is the case when the lengths of finite elements are small. In such case, the coefficient matrix of Eq. (33) can be hold during Newton-Raphson iterations, which significantly reduces the computation time. However, the approximation technique may not converge for a system whose is large. To overcome this numerical difficulty, a combined incremental and iterative method (see Ref. 16) can be used.

q

q

5.4. NUMERICAL ALGORITHM

Kinematics of the relative nodal displacements and the equations of equilibrium are presented in the section 3. This section explains how the equations are implemented to obtain the relative and absolute nodal displacements of a structure. The numerical algorithm for closed loop systems is as follows:

1) Perform the graph theoretic preprocessing to determine computational path

sequences. 2) Form a stiffness matrix . K3) Compute Φ , , and for in the backward path sequence. qΦ *Q kq

4) Solve the Eq. (37) to obtain and q∆ λ∆ . 5) If and F q∆ remains within the specified allowable error tolerance, then

go to step 6. Otherwise, improve the solution using Eq. (35). Go to step 3. 6) Compute the Cartesian deformations in the forward path sequence by using

Eqs. (1) and (2).




Static analysis of a cantilever beam subjected to end moment M , as shown in

Fig. 6 is carried out.

M

X

Y

][0.1][0.1][0.12

0.0]/[100.3

4

2

27

mImAmL

mNE

====

×=ν

L

Figure 6 A cantilever beam subjected to end moment

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 13-2

0

2

4

6

8

10

12

14

16

Undeformed Proposed ANSYS: nonlinear ANSYS: linear

Y [m

]

X [m]

Figure 7 Deformed shape of the beam

5-13

In the figure, E , ν , L , , and denotes Young's modulus, Poisson ratio, the length of the beam, the cross sectional area of the beam, and the second area moment of the cross section, respectively. M =6.545×106 [N·m] is applied at the end node. Fig. 7 shows the deformed shapes of the beam by the proposed method by the proposed method and a commercial program ANSYS. In the figure, Proposed, ANSYS: nonlinear, and ANSYS: linear denote numerical results by the proposed method, a commercial program ANSYS using nonlinear analysis, and ANSYS using linear analysis, respectively. It shows that the numerical results obtained by the proposed method and ANSYS(nonlinear analysis) are almost identical, but the numerical results by ANSYS(linear analysis) shows large difference with the remaining two numerical results.

A I

0 2 4 6 8 10

-10

-8

-6

-4

-2

0

ue: Proposed ue: ANSYS

u e [m

]

The number of elements

Figure 8 Convergence of axial deformation at the end node vs. the number of elements

Fig. 8 shows the convergence of the axial deformation at the end node. When fewer elements are used for static analysis, the numerical results of ANSYS are more accurate than those of the proposed method, but obtained

eu

eu



by the proposed method converges rapidly as the number of elements is increased. In the figure, the numerical results with more than 6 elements by the two methods are almost identical. From the analysis results, it is known that the effect of the nonlinear stiffness matrix is diminished rapidly as the number of elements is increased.

-1 0 1 2 3 4 5 6 7 8

0

1

2

3

4(F = [3.0E+4, -1.0E+04 ]T [N], M= 0.0E+0[Nm])

Undeformed shape Deformed shape: 20 elements Deformed shape: ANSYS

Y [m

]

X [m]

F

Y

M

X

][002.0][01.0][14.14

0.0]/[100.3

4

2

27

mImAmL

mNE

====

×=ν

4π

4π

L

P

Figure 9 A closed loop system subjected to concentrated force and Moment

Figure 10 Comparison of deformed shapes of the closed system

5-15

Fig. 9 shows a closed loop system subjected to a concentrated force and moment

FM at a point P . When =[3×10F 4 -1×104]T [N] and M =0.0

[N·m] are applied at the point P the deformed shapes of the system are shown in Fig. 10. It shows that the numerical results obtained by the proposed method with 20 elements and a commercial program ANSYS are almost identical.

-1 0 1 2 3 4 5 6 7 8-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0(F=[3.0E+4, -3.0E+4]T[N], M=3.0E+4[Nm])

Y[m

]

X[m]

Undeformed shape Deformed shape: 20 elements

Figure 11 Undeformed and deformed shapes of the closed loop system

0 1 2 3 4 5 6 7 8-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

(F=[3.0E+4, -3.0E+4]T[N], M=3.0E+4[Nm])

Y[m

]

X[m]

0.33F, 0.33M 0.67F, 0.67M 1.00F, 1.00M

Figure 12 Deformed shapes of the closed loop system at each load step



When =[3×10F 4 -3×104]T and M =3.0×104[N·m] are applied at the point P , the deformed shape of the system is shown in Fig. 11. While the numerical solution by the proposed method converges after the 7-th iteration, that by the commercial program ANSYS does not converge. Fig. 12 shows the deformed shape of the system at each load step.

5.6. CONCLUSIONS

A geometric nonlinear formulation for structures undergoing large deformations is investigated in this research. Nodal displacements in the proposed method are referred to its adjacent nodal reference frame. Since the nodal displacements are measured relative to its inboard nodal frame, quantity of the nodal displacements is still small for a structure undergoing large deformations if the element sizes are small. Relative coordinate kinematics is developed to define relative position and orientation of the nodal displacements. As a consequence, many element formulations developed under small deformation assumptions are reusable for structures undergoing large deformations, which makes it easy to develop a computer program. A structural system is represented by a graph to systematically develop the governing equations of equilibrium for general systems. Closed loops are opened to form a tree topology by cutting nodes. Two computational sequences are defined for a graph. One is the forward path sequence that is used to recover the Cartesian nodal deformations from relative nodal displacements and traverses a graph from the base node towards the terminal nodes. The other is the backward path sequence that is used to recover the nodal forces in the relative coordinate system from the known nodal forces in the absolute coordinate system and traverses from the terminal nodes toward the base node. A solution algorithm is developed to implement the proposed method. Static analyses are performed for structures undergoing large deformations. The proposed method can solve the problem which cannot be solved by the commercial program ANSYS.

5-17

REFERENCES

(1) El Damatty, A. A., Korol, R. M and Mirza, F. A., "Large Displacement Extension of

Consistent Shell Element for Static and Dynamic Analysis," Computers & Structures, Vol.

62, No. 6, (1997), p. 943-960.

(2) Mayo, J and Domínquez, J., "A Finite Element Geometrically Nonlinear Dynamic

Formulation of Flexible Multibody Systems using a New Displacements

Representation," J. Vibration and Acoustics, Vol. 119, (1997), p.573-580.

(3) Dhatt, G and Touzot, G., The Finite Element Method Displayed, John Wiley & Sons,

(1984).

(4) Bathe, K. J., Finite Element Procedures, Prentice-Hall, (1996).

(5) Avello, A. J., Jolón, G. D. and Bayo, E., "Dynamics of Flexible Multibody Systems

using Cartesian Co-ordinates and Large Displacement Theory," Int. J. Numer. Methods

Eng., Vol. 32, No. 8, (1991), p.1543-1564.

(6) Shabana, A. A, "An Absolute Nodal Co-ordinate Formulation for the Large Rotation

and Deformation Analysis of Flexible Bodies," Technical Report MBS 96-1-UIC,

Department of Mechanical Engineering, University of Illionois at Chicago, (1996).

(7) Shabana, A. A. and Christensen, A., "Three Dimensional Absolute nodal coordinate

formulation: Plate Problem," Int. J. Nuner. Methods Eng., Vol. 40, No. 15, (1997),

p.2275-2790.

(8) Shabana, A. A., Dynamics of Multibody Systems, 2nd edition, Cambridge University

Press, (1998).

(9) Takahashi, Y. and Shimizu, N, “Study on Elastic Forces of the Absolute Nodal

Coordinate Formulation for Deformable Beams,” Proceedings of the ASME Design

Engineering Technical Conferences, (1999).

(10) Featherstone, R., "The Calculation of Robot Dynamics Using Articulated-Body

Inertias, " Int. J. Roboics Res., Vol. 2, (1983), p. 13-30.

(11) Bae, D. S. and Haug, E. J., "A Recursive Formulation for Constrained Mechanical

System Dynamics: Part I. Open Loop Systems," Mech. Struct. and Machines, Vol. 15, No.



3, (1987), p. 359-382.

(12) Bae, D. S. and Haug, E. J., "A Recursive Formulation for Constrained Mechanical

System Dynamics: Part II. Closed Loop Systems," Mech. Struct. and Machines, Vol. 15,

No. 4, (1987), p. 481-506.

(13) Lin, T. C. and Yae, K. H., Recursive Linearization of Multibody Dynamics and

Application to Control Design, Technical Report R-75, Center for Simulation and


Mathematics, The University of Iowa, Iowa City, Iowa, (1990).

(14) Yoo, H., Ryan, R. and Scott, R., "Dynamics of Flexible Beams undergoing Overall

Motion," J. Sound and Vibration, Vol. 181, No. 2, (1995), p.261-278.

(15) Haug, E. J., Computer-Aided Kinematics and Dynamics of Mechanical Systems:

Volume I. Basic Methods, Allyn and Bacon, (1989).

(16) Crisfield, M. A., Non-Linear Finite Element Analysis of Solids and Structures, Wiley,

(1997).

5-19



6

DYNAMIC ANALYSIS OF HIGH-MOBILITY TRACKED VEHICLES

6.1. INTRODUCTION

High-speed, high-mobility tracked vehicles are subjected to impulsive dynamic loads resulting from the interaction of the track chains with the vehicle components and the ground. These dynamic loads can have an adverse effect on the vehicle performance and can cause high stress levels that limit the operational life of the vehicle components. For this reason, high speed, high mobility tracked vehicles have sophisticated suspension systems, a more elaborate and detailed design of the links of the track chains, and improved vibration characteristics that allow the vehicle to perform efficiently in hostile operating environments.

Galaitsis [1] demonstrated that the predicted dynamic track tension and suspension loads in a high speed tracked vehicle developed by an analytical method are useful in evaluating the dynamic characteristics of the tracked vehicle components. The predicted track tension was compared with the measured data from a military tracked vehicle. Bando et al [2] developed a planar computer model for rubber tracked bulldozers. Steel and fiber molded continuous rubber track is discretized into several rigid bodies connected by compliant force elements. Characteristics of track damage, vibration, and noise are investigated using the simulation results. Nakanishi and Shabana [3] developed a two-dimensional contact force model for planar analysis of multibody tracked vehicle systems. The stiffness and damping coefficients in this contact force model were determined based on experimental observations of the overall vibration characteristics of the tracked vehicle. The nonlinear equations of motion of the vehicle were obtained using the Lagrangian approach and the algebraic constraint equations that describe the joints and specified



motion trajectories are adjoined to the system equations of motion using the technique of Lagrange multipliers. The generalized contact forces associated with the system generalized co-ordinates were obtained using the virtual work. Choi [4] presented a large-scaled multibody dynamic model of construction tracked vehicle in which the track is assumed to consist of track links connected by single degree of freedom pin joints. In this detailed three-dimensional dynamic model, each track link, sprocket, roller, and idler is considered as a rigid body that has a relative rotational degree of freedom. Scholar and Perkins [5] developed an efficient alternative model of the track chains considering longitudinal vibrations. The track is assumed to consist of a finite number of segments and each is modeled as a continuous uniform elastic rod attached to the vehicle wheels. Each segment consists of several track links which are collectively lumped as a single body so that overall chain stretching effects are accounted for.

A detailed three-dimensional tracked vehicle model as the one developed by Choi [4] may have hundreds or thousands differential and algebraic equations. These equations are highly non-linear and can only be solved using matrix, numerical and computer methods. In addition to this dimensionality problem, tracked vehicles are characterized by impulsive forces due to the contacts between the track links and the vehicle components as well as the ground. The impulsive contact forces cause serious numerical problems when the vehicle equations of motion are integrated numerically. The degree of difficulty may significantly increase if compliant elements, instead of the ideal pin joints, are used to model the connection between the links of the track chains, as it is the case in this investigation. The compliant elements must have very high stiffness coefficients in order to maintain the link connectivity. These stiffness coefficients, which are determined experimentally in this investigation, introduce high frequency oscillatory components to the solution, thereby forcing the numerical integration routine to take a very small time step size. It is, therefore, important to adopt a numerical scheme that can be efficiently used in modeling this type of vehicle. Newmark [6] presented an absolutely stable second-order numerical integrator in the area of structural dynamics. The Newmark integrator was modified by Wilson [13] so that highly oscillatory state variables are numerically damped out. The numerical damping algorithms are extended

6-3

and generalized in implicit and explicit forms with a constant step size by Chung [7,8]. The algorithm developed by Chung is employed in this investigation due to its easy implementation and large stability region.

The objective of this investigation is to develop a computational procedure for the nonlinear dynamics of high-speed, high-mobility tracked vehicles. The model developed in this investigation differs from the low-speed tracked vehicle model previously developed by Choi [4] in two important aspects summarized as follows:

(1) The high speed tracked vehicle considered in this investigation has a

sophisticated suspension system that consists of road arms and wheels instead of the simple roller type suspension system previously developed by Choi.

(2) In the model previously developed by Choi [4], the links of the track chains are connected by pin joints that have one degree of freedom. In the model developed in this investigation, compliant force elements are used to model the connectivity between the links of the track chains. The characteristics of these compliant elements are determined experimentally as discussed in Section 5.

The application of the numerical integration scheme developed by Chung [7,8] to tracked vehicle dynamics is investigated in this paper using different simulations scenarios that include accelerated motion, high speed motion, braking and turning motion.

6.2. HIGH-SPEED, HIGH-MOBILITY TRACKED VEHICLES

In this section, the high-speed, high-mobility tracked vehicle model used in this investigation is described. The three-dimensional model, which is shown in Fig. 1, represents the third generation of a military vehicle weighing approximately 50 tons and can be driven at a speed higher than 60 km/h.



Figure 1. High mobility tracked vehicle.

The vehicle consists of a chassis and two track subsystems. The chassis subsystem includes a chassis, sprockets, support rollers, idlers, road arms, road wheels and the suspension units. The sprockets, support rollers, and road arms are connected to the chassis by revolute joints. The suspension unit includes a Hydro-pneumatic Suspension Unit (HSU)[17], and torsion bar that are modeled as force elements whose compliance characteristics are evaluated using analytical and empirical methods. The HSU systems are mounted on front and rear stations to damp out pitching motion and to decrease the vehicle speed when the vehicle is running over large obstacles. The spring torque of the HSU systems can be written as

1HSU PALT = (1)

where P is the gas pressure, is the area of piston, and is the distance shown in Fig. 2. The pressure

A 1LP in the gas chamber of HSU system with

respect to rotation angle of a road arm is defined as

γ

−+

=)( 22 LLl

lPPis

ii (2)

where , , and are the initial pressure and distances when the road arm

is in its initial configuration, iP il iL2

γ is a constant which is equal to 1.4, and is the distance shown in Fig. 2.

2L

6-5

Figure 2. Schematic diagram of spring-damper suspension units: hydro pneumatic

suspension unit and torsion bar systems.

The distance l can be adjusted by charging or discharging oil into the oil chamber. The torsion bars are mounted on the middle stations for this vehicle model. A simple torsional spring model is used in this investigation to represent the stiffness of the torsional bars. The stiffness coefficient of the torsion bar spring is approximately Nm/rad. Figure 2 shows the schematic diagram of the HSU and the torsion bar systems. Figure 3 shows the spring characteristics which are employed in this investigation.

s

4105×

Figure 3. Spring characteristics of suspension unit

Each track subsystem is modeled as a series of bodies connected by rubber

bushings around the link pins which are inserted into a shoe plate with some radial pressure in order to reduce the non-linear effect of the rubber. When the



vehicle runs over rough surfaces, the track chains are subjected to extremely high impulsive contact forces as the result of their interaction with the vehicle components such as road wheels, idlers, and sprocket teeth, as well as the ground. The rubber bushings and double pins tend to reduce the high impulsive contact forces by providing cushion and reducing the relative angle between the track links. About 10 percent of the vehicle weight is given as the pre-tension for the track to prevent frequent separations of the track when the vehicle runs at a high speed. About 14 degrees of a pre-torsion is also provided in order to reduce the fluctuation of the torque in the rubber bushing when the track links contact the sprocket and idler.

The vehicle, which presents a high-mobility military tracked vehicle, consists of one hundred eighty nine bodies; body 1 is the chassis, bodies 2 and 3 are the right and left driving sprockets, bodies 4 - 17 are the right and left arms, bodies 18 - 31 are the right and left wheels, bodies 32 - 37 are the right and left support rollers, and bodies 38 - 113 and 114 - 189 are the right and left track links, respectively. The sprockets, rollers and arms are connected with chassis by 22 revolute joints, and wheels are connected with arms by 14 revolute joints, each of which has one degree of freedom. This vehicle model has 152 bushing elements between track links, and 954 degrees of freedom.

6.3. KINEMATIC RELATIONSHIPS AND EQUATIONS OF MOTION

In this investigation, the relative generalized co-ordinates are employed in order to reduce the number of equations of motion and to avoid the difficulty associated with the solution of differential and algebraic equations. Since the track chains interact with the chassis components through contact forces and since adjacent track links are connected by compliant force elements, each link in the track chain has six degrees of freedom which are represented by three translational co-ordinates and three Euler angles [9]. Recursive kinematic equations of tracked vehicles were presented by Choi [4,16], who showed that the relationship between the absolute Cartesian velocities of the chassis components can be expressed in terms of the independent joint velocities as

6-7

riqB &=q& (3)

where , , and q are relative independent co-ordinates, velocity transformation matrix, and Cartesian velocities of the chassis subsystem, respectively. The equations of motion of the chassis that employs the velocity transformation defined in the preceding equations are given as follows:

riq B

)qBMQBqMBB r

iTr

iT ( &&&& −= (4)

where is the mass matrix, and is the generalized external force vector of the chassis subsystem. Since there is no kinematic coupling between the chassis subsystem and the track subsystems, the equations of motion of the chassis subsystem can be obtained using the preceding equation as follows:

M Q

Ci

ri

Ci QqM =&& (5)

where , . MBBM TC

i = )( ri

TCi qBMQBQ &&−=

For the track subsystems, the equations of motion can be written as

ttt QqM =&& (5)

where , and denote the mass matrix; and the generalized coordinate and force vectors for the track subsystem, respectively. Consequently, the accelerations of the chassis and the track links can obtained by solving Equations (5) and (6).

tM tq tQ

6.4. A COMPLIANT TRACK MODEL


Two models can be used to connect the track links of the high-mobility tracked vehicle chains. These two models are shown in Fig. 4. In the first model, shown in Fig. 4(a), a single pin is used to connect two links of the chain. In the


second model, shown in Fig. 4(b), two pins are used to connect the track links. In both models, rubber bushings are inserted between the pins and the track links, and as a consequence, the relative rotations between the pins and the links are relatively small. In this section, the force models used for the single pin and double pin connections are described.

(a) Single pin track links (b) Double pin track links

Figure 4. Track links of high mobility tracked vehicle

6.4.1 SINGLE PIN CONNECTION

Figure 5. Single pin connection

6-9

Figure 5 shows the details of the link, pin and bushings connection of a single

pin track link. In this investigation, a continuous force model is used to define the pin joint connections. This force model is a non-linear function of the co-ordinates of the two links. In order to define the generalized compliant bushing forces, several coordinate systems are introduced. Two centroidal body coordinate systems and for the track links and i

bi

bib ZYX j

bj

bj

b ZYX i j ,

respectively; a joint coordinate system iii ZYX whose origin is assumed to be located at the geometric center of the circular groove containing the pin and the bushing; and a pin coordinate system jjj ZYX whose origin is rigidly attached to the center of the pin. Note that because of the bushing effect, the origins of the joint and pin coordinate systems do not always coincide. The displacement of the pin coordinate system jjj ZYX with respect to the joint coordinate system

iii ZYX is a function of the bushing stiffness. Also note that the location and orientation of the joint coordinate system iii ZYX can be determined as a function of the generalized co-ordinates of link . For simplicity, it is assumed in this investigation that the location and orientation of the pin coordinate system can be defined in terms of the co-ordinates of link

i

j . The deviation

shown in Figure 5 can be used to determine the generalized

forces acting on the two links and

Tzy ],, δδxR [δ=δ

i j as the result of the bushing effect. The bushing force and torque applied to the frame j are given as follows:

−

=

θθθθθ δδ

00C

δδ

KK

QQ

&

&RRRR

j

jR

C00

where and are the 3 RR ,, CKK θ θC × 3 diagonal matrices that contain the

stiffness and damping coefficients of the bushing, and is translational force vector and is the vector of translational deformations of the frame

jRQ

Rδ j relative

to the frame . Similarly, is the rotational force vector and δ is the

vector of relative rotational deformations of the frame

i jθQ ϑ

j relative to the frame . The force and torque applied to the frame i are assumed to be equal in

i



magnitude and opposite in direction to the force and torque acting on frame j . Once these forces are determined, the generalized bushing forces associated with the generalized co-ordinates of the track links and i j can be determined.

6.4.2 DOUBLE PIN CONNECTION

Figure6. Double pin connection

In the double pin assembly, shown in Figure 6, two adjacent track links are

connected with a connector element using two pins and rubber bushings. The mass and mass moment of inertia of the connector element are relatively small as compared to those of the track links. Therefore, the dynamic effects of connector element are modeled in this investigation. This approach has the advantage of reducing the number of degrees of freedom of the system. The double pin assembly can be modeled by considering one radial, one axial, and three rotational springs. The radial spring provides the restoring force due to the combined translational deformation of the two rubber bushings along the radial direction of the connector as shown in Fig. 6. The axial spring restricts the translational motion of the two links along the lateral direction as shown in Fig. 6. The rotational springs are used to model the relative rotational deformation

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between the two track links. The length of the radial spring is assumed to be the distance between the origins of the coordinate systems

liii ZYX and

jjj ZYX shown in Fig. 6. This distance is defined as ( ) ijij2 ddl

T= (8)

The magnitude of the force produced by the radial spring is

( ) lCllKF r0rr&+−= (9)

where is the spring stiffness coefficient, and is the damping coefficient, and is obtained by differentiating Equation (8) with respect to time. Similarly, the restoring force due to the translational spring along the

rK rC

iiZ axis is

i,j

zRzi,j

zRzz CKF δ−δ−= & (10)

where is translational deformation of the i,jzδ

jjj ZYX frame with respect to

the iii ZYX frame along the Z axis, and are the stiffness and damping coefficients.

RzK RzC

The first two components of the bushing restoring torque as the result of the relative rotation of link with respect to link i j are given by

i,j

xxi,j

xxx CKT θθ θθ&−−= (11)

i,jyy

i,jyyy CKT θθ θθ

&−−= (12)

where and are relative rotational deformations of the i,jxθ

i,jyθ

jjj ZYX frame

about x-axis and y-axis with respect to the iii ZYX frame, respectively, , , , and are stiffness and damping coefficients. The restoring bushing

torque about the

xKθ

yKθ xθC yCθ

jZ axis due to the rotation of link j with respect to link i

ib,j

zzib,j

zzz CKT θθ θθ&−−= (13)



where is relative angle between the link and the connector element and can be obtained by defining the components of the in the coordinate

system of link as

ib,jzθ i

ijd i

i

iji dAdT

=

=ijz

ijy

ijx

ij

ddd

(14)

It follows that ( )ij

xijy

ib,iz d/d-1tan=θ (15)

where is the transformation matrix that defines the orientation of link with respect to the global frame. Note that since the inertia of the connector element is neglected, the resultant force acting on this element must be equal to zero. Using the spring forces defined in this section, the generalized bushing forces acting on the track links can be systematically defined.

iA i

6.5. MEASUREMENT OF TRACK COMPLIANCE CHARACTERISTICS

In order to determine the stiffness and damping coefficients of the contact force models used in this investigation, an experimental study is conducted to examine the road wheel and track link contact as well as the interaction between the track links. Since the experimental results are to be used in the dynamic simulation of the multibody tracked vehicle, the dynamics of the contact is also considered in the measurement process.

While a viscous damping force is proportional to the velocity, in many cases, analytic expressions for the damping forces are not directly available. It is, however, possible to obtain an equivalent viscous damping coefficient by equating energy expressions before and after a contact. In this investigation, the effective stiffness and damping coefficient are obtained by employing the hysteresis loop method [10]. The effective stiffness and damping coefficient of single degree of freedom system are given as follows [10]:

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φ+ω= cos

xFm

0

02effeffK (16)

ωφsin

0

0

xF

Ceff = (17)

where , , and are the effective mass, the magnitude of applied force, the magnitude of displacement, the natural frequency, and the phase angle of displacement, respectively.

effm , 0F 0x , ω φ

In these experimental studies, forces are applied to the center of the road wheel which is in contact with a track link fixed to a rigid frame. A LVDT sensor is attached between the center of the wheel and a track link fixed base to measure the relative displacement. For a static test, the actuator force is increased gradually up to 10 ton with 2mm/min velocity. For a dynamic test, the actuator force is excited harmonically up to 35 Hz. Frequencies higher than 35 Hz are not considered in the measurement because of noise and system resonance. The relationship between the effective stiffness, damping coefficient, and frequencies is given by Park et al [11]. It can be shown that the effective stiffness increases up to a frequency of 10 Hz and does not significantly change after this frequency. On the other hand, the effective damping coefficient decreases as the frequency increases.

A LVDT sensor is attached between two adjacent track links to measure the relative displacement. For a static test, the actuator force is increased gradually up to 10 ton with 2mm/min velocity. Figure 7 shows the resulting load-displacement relationship. For a dynamic test, a harmonic actuator force with a frequency up to 50 Hz is used. Figure 8 shows the hysteresis loop when the load frequency is 10 Hz with 5 ton pre-static applied force. It can be observed that the effective stiffness increases up to 12 Hz and does not significantly change after this frequency. The effective damping coefficient, on the other hand, decreases as the frequency increases.



Figure 7. Load-displacement relationship for Radial static measurement

(Pre-static load: 5ton, Forcing freq.: 10HZ)

Figure 8. Hysteresis loop for radial dynamic test

A connector end is welded to the fixed track shoe plate and the other end of

the connector is attached to a load cell, which is connected to an actuator cylinder by revolute joint. Fourteen degrees of the pre-set angle is given in the pin. A torque of 500 ton-mm is applied along the directions of rotation. The

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static torque versus the rotational angle, the effective torsional stiffness versus frequency, and the effective damping coefficient versus frequency are plotted by Park et al [11]. The experimental results showed that the effective torsional stiffness is less sensitive to the loading frequency and the effective damping coefficient decreases to a small value when the frequency exceeds 20 Hz.

In this investigation, for the sake of simplicity, the stiffness and damping coefficients used in the force models are determined using empirical methods based on the results of the static test only. A spline curve fitting is used to obtain the compliant characteristics between measurements.

6.6. CONTACT FORCES

In this section, the methods used for developing the contact force models used in this investigation are briefly discussed. The scenarios of the contacts between the track links and the road wheels, rollers, sprockets, and the ground are explained. A more detailed discussion on the formulation of the contact forces is presented by Choi, et al, [4, 12], and Nakanishi and Shabana [3].

(a) inner surface contact (b) edge contact

Figure 9. Track link and wheel interactions

6.6.1 INTERACTION BETWEEN TRACK AND ROAD WHEEL, IDLER, AND SUPPORT ROLLER

As shown in Fig. 9, each roller of the vehicle model used in this investigation



consists of two wheels which are rigidly connected. There are four different possibilities for the roller and track interaction. The first possibility occurs when a track link and one wheel of the roller are in contact. In this case, a concentrated contact force is used at the center of the contact surface of the wheel. The contact force acting on the link is assumed to be equal in magnitude and opposite in direction to the force acting on the roller. The second possibility occurs when both wheels of the roller are in contact with the track link. In this case, two concentrated contact forces are applied to the roller and the track link. The third and fourth possibilities occur, respectively, when either one wheel or both wheels are in contact with the edges of track link. In such a case, one or two concentrated contact forces are applied to the wheel and the edge of the track link.

6.6.2 TRACK CENTER GUIDE AND ROAD WHEEL INTERACTIONS

(a) side wall contact (b) top surface contact

Figure 10. Center guide and wheel interactions

Figure 10 shows a schematic diagram for a track center guide and a road wheel when they are in contact. As previously pointed out a road wheel of the vehicle model used in this investigation consists of two wheels, which are rigidly connected, and therefore, there are four possibilities for the track center guide and wheel interactions, as shown in the Figure 10. The first possibility is the case in which the right side plate of the wheel is in contact with the left side wall of

6-17

the track center guide. In the second possibility, the left side plate of the wheel is in contact with the right side wall of the track center guide. The third possibility occurs when one bottom surface of wheel and the top surface of track center guide are in contact. In these three contact cases, a concentrated contact force is introduced at the contact surface of the road wheel, and that contact force is equal in magnitude and opposite in direction to the force acting on the track link. The fourth possibility occurs when the two road wheels are not in contact with the track center guide. In this case, no generalized contact forces will be introduced.

6.6.3 INTERACTION BETWEEN THE SPROCKET TEETH AND TRACK LINK PINS

In this investigation, five tooth surfaces are used to represent the spatial contact between the sprocket teeth and the track link pins. During the course of engagement between the sprocket teeth and the track links, several sprocket teeth can be in contact with several track link pins, as shown in Fig. 11. The sprocket used in this investigation has ten teeth, and each tooth has five contact surfaces. These surfaces are the top, the left, the right, front, and back surfaces. A Cartesian coordinate system is introduced for each surface. The surface coordinate system is assumed to have a constant orientation with respect to a selected tooth coordinate system. The tooth coordinate system has a constant orientation with respect to the sprocket coordinate system. Therefore, the orientation of a surface coordinate system can be defined in the global system using three coordinate transformation matrices; two of them are constant and the third is the time dependent rotation matrix of the sprocket. Using these coordinate transformations and the absolute Cartesian co-ordinates of the origin of the sprocket coordinate system, the location and orientation of each tooth surface can be defined in the global coordinate system. Using the track link coordinate system, the global position vector of the center of the track link pin can be defined.



(a) sprocket teeth and connector contact (b) teeth side wall and link side wall

contact

(c) teeth top surface and link inner surface contact

Figure 11. Sprocket tooth and track interaction

This vector and the global co-ordinates of the tooth surfaces can be used to determine the position of the track link pins with respect to the sprocket teeth. The relative position of the track link pins, with respect to the sprocket teeth can be used to develop a computer algorithm that determines whether or not the track link pin is in contact with one of surfaces of the sprocket teeth. The interactions between the track link pins and the sprocket base circle are also considered in this investigation. To this end, the distance between the center of the track link pin and the center of sprocket is monitored. When this distance is less than the sum of the pin radius and the sprocket base circle radius, contact is assumed and a concentrated force is applied to the sprocket and the track link pin.

6.6.4 GROUND AND TRACK SHOE INTERACTIONS

The track link used in this investigation has a single or double shoe plate, and

therefore, there are one or two surfaces on each track link that can come into contact with the ground. The global position vectors that define the location of points on the shoe plates are expressed in terms of the generalized co-ordinates of the track links and are used to predict whether or not the track link is in contact with the ground. In this investigation, contact forces are applied at selected six points on the track link shoe when it comes into contact with the

6-19

ground. The normal force components are used with the coefficient of friction to define the tangential friction forces [4, 12].

6.7. METHOD OF NUMERICAL INTEGRATION

The equations of motion of a tracked vehicle are formulated as a set of differential equations, as described in Section 3. The solution of the differential equations can be obtained by step-by-step numerical integration. There are two types of integration methods; one is the implicit method and the other is the explicit method. The implicit method generally has a larger stability region, but it requires solving a system of nonlinear equations. The explicit method, on the other hand, has relatively smaller stability region, but it requires solving only a system of linear equations. In this investigation, an explicit method is employed.

The dynamics of tracked vehicles is characterized by high impulsive forces resulting from the contact between the track chains and the vehicle components as well as the ground. Because of the high frequency impulsive forces, the numerical integration routine is forced to take a small time step, and as a consequence, the simulation of a complex tracked vehicle model, as the one described in this paper, represents a challenging task. Nonetheless the high frequency oscillations may have little influence on the low frequency motion. In this case, the high oscillations can be damped out to obtain the gross motion of the track link. Various dissipation algorithms for time integration of structural systems have been proposed [7,8,13]. In this investigation, the method proposed by Chung and Lee [7] is considered because of its easy implementation and computational efficiency. Accuracy and stability conditions must be considered in carrying out a numerical integration of the tracked vehicle equations. The accuracy and stability conditions are obtained by using the truncation error and the error propagation analyses. A variable step algorithm is proposed in the following subsections.

6.7.1 ACCURACY ANALYSIS

The following numerical integrator proposed by Chung and Lee [13] is



employed in this research.

)q,N(qMq 1nnn &&& −= (18)

])2/3()2/1[(∆ n1nn1n qqqq &&&&&& +−+= −+ t (19) ])27/28()54/29[( n1nnn1n qqqqq 2 &&&&& +−∆+∆+= −+ tt (20)

The can be expanded by the Taylor series as follows: 1n+q

)(O)2/( 3n

2nn1n ttt ∆+∆+∆+=+ qqqq &&& (21)

where is collection of higher order terms. Subtracting Equation (20) from Equation (21) yields the truncation error as follows:

)(O 3t∆

)(O))(54/29()( 32

n1nn ttt ∆+∆−=τ − qq &&&& (22)

)(Odtd)54/29( 33

n tt ∆+∆≈ q&& (23)

which shows that the proposed integrator achieves the second-order accuracy for non-linear dynamic systems.

6.7.2 STABILITY ANALYSIS

Since it is difficult to analyze the stability condition for a general nonlinear

system, the following linear, undamped, and unloaded system is considered:

02 =+ nn qq ω&& (24)

where is a natural frequency. Applying Equation (24) with the integration formula proposed in this section yields the one step form of the numerical scheme

ω

nn HXX =+1 1,......,2,1,0 −∈ Nn (25) where

Tn

2nnn ]q,q,q[ &&& tt ∆∆=X (26)

and

6-21

Ω−−Ω−

−Ω−=

002/11)2/3(54/291)27/28(1

2

2

2

H (27)

in which t∆ω=Ω . The characteristic equation for is obtained as follows: H

0)27/1()27/2(1)27/28(2)det( 2223 =Ω+Ω−+Ω−−=−− λλλλIH (28)

where is the I 33× identity matrix and λ denotes the eigenvalue. The stability characteristics of the method are determined by the condition that the roots of the characteristic equation remain in or on the unit circle of the complex plane as follows:

1≤ρ , 321 ,,max λλλ=ρ (29)

where is called the spectral radius. Stability analysis can be assessed by using the transformation of Eq. 7.9 to map the interior of the unit circle into the left half-plane and by applying the Routh-Hurwitz criteria to the transformed characteristic equation. The stability condition for the algorithm is obtained by applying the Routh-Hurwitz criteria as follows:

ρ

0)27/31(4 2 ≥Ω− (30) 0≥)27/23(4 2Ω− (31)

which are reduced to

)/8665.1( ω≤∆t (32)

Equation (32) provides a guideline in choosing a step size that satisfies the stability condition.



6.7.3 IMPLEMENTATION OF A VARIABLE STEPPING ALGORITHM

Since the governing equations of motion for the tracked vehicle system are highly nonlinear, the integration step size must be varied so that both the accuracy and stability conditions are satisfied. For the accuracy condition, ignoring the higher order terms in Equation (22) yields the local truncation error formula as follows:

2n1nn |)(|)54/29()( tt ∆−= − qqτ &&&& (33)

The allowable stepsize with a given error tolerance τ is obtained by solving

Equation (33) for t∆ as follows:

2/1n1n ||)(|29/54| qqτ &&&& −=∆ −t (34)

For the stability condition, the apparent frequency method proposed by Park

and Underwood [14] is employed in this research. An apparent frequency is estimated by substituting q and into the following equation: q&&

01

21 =∆+∆ ++

inapp

in qq ω&& i (35) ,......,2,1,0 qn∈

where is the apparent frequency and is the number of generalized co-

ordinates. The highest apparent frequency is selected as the reference frequency in determining the step size.

appω qn

The step size determination algorithm is shown in Figure 12. Note that the stability condition of instead of in Equation

(32) is used for conservative numerical integration. The integration step size employed by the variable step integration algorithm used in this investigation, when the vehicle maximum acceleration, steady state velocity at 50 Km/h and stiff deceleration of braking, is shown in Figure 13. This figure shows that the integration step size is relatively depended on the vehicle speed. The increment of vehicle speed will enlarge impulsive contact forces and oscillation of track links, and integration step size should be decreased, accordingly.

app/ω5.0<∆t app/ω8665.1<∆t

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Figure 12. Variable stepsize algorithm

Figure 13. Stepsize of variable step integration algorithm




The high mobility tracked vehicle shown in Figure1 is used as a simulation model in order to demonstrate the use of the methods proposed in this paper. Several simulation scenarios, including acceleration, high speed motion, braking and turning motion, are presented in this investigation. In the simulation of acceleration, high speed motion, and braking of the vehicle, the same angular velocity is used for both left and right sprockets in order to obtain straight line motion. The angular velocities of the sprockets are increased linearly up to -45 rad/s in 10 s, kept constant for 3 s, and then decreased linearly to 0 rad/s in 4 s. The coefficient of friction between the track links and ground is assumed to be 0.7 in the case of rubber and concrete contact. The double pin track link is used in the numerical study presented in this section. Figures 14-18 show the numerical results of simulation of the acceleration, steady-state velocity and deceleration.

Figure 14. Vertical motion of a track link

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Figure 15. Radial tension of track link

The vertical displacement of a track link with respect to the global coordinate system during the constant velocity motion is shown in Figure 14. This figure clearly shows the effect of three support rollers, idler, six road wheels and sprocket on the vertical displacement of the track link. The track tension can have a significant effect on the dynamic behavior of tracked vehicles, such as preventing the separation of the track chains [18,19] from chassis, distribution of mean maximum pressure(MMP), power efficiency, and the life of the track chain. As previously pointed out in Section 2, about 10 percent of the vehicle weight [15] is used as track pre-tension. Simulation results showed that the track tension significantly decreases after the start of the motion.

(a) (b)



(c) (d)

(e)

Figure 16. Sprocket teeth loading contour: (a) acceleration; (b) cruise at high speed; (c) braking; (d) turning (right sprocket); (e) turning ( left sprocket)

Figure 15 shows the longitudinal track tension in the bushing between track links, while Figure 16 shows sprocket teeth loading contour. Heavy duty high mobility tracked vehicles as one used in this study have, in general, double pin track links. One of the main advantages of using double pin track is that the shear stress on the rubber bushings can be significantly reduced as compared to the single pin track link. In order to compare the loadings on the track bushings in the case of single or double pins, new driving conditions are examined. The rotational speed of both sprockets is decreased linearly up to - 9 rad/sec in 2 sec, and then kept constant velocities.

Figure 17 illustrates the moment on the rubber bushings in the case of the single and double pin track. The results presented in this figure demonstrate the significant reduction of the load on the rubber bushings when a double pin track is used. Figure 18 shows the norm of the contact forces exerted on one of the links of the right track chain as the result of its interaction with the road wheels,

6-27

support rollers, idler, sprocket, and ground. Figure 19 shows a road arm angle and HSU gas pressure of the second road wheel.

Figure 17. Torsional moments of track rubber bushing.

Figure 18. Contact forces of track link



Figure 19. Gas pressure of HSU system

The second simulation scenario used in this study is a turning motion. The

turning motion is obtained by providing two different values for the angular velocities of the sprockets. The angular velocity of the right sprocket is decreased linearly to -9 rad/s and the angular velocity of the left sprocket is increased linearly to 9 rad/s in 2 s. The angular velocities are then kept constant velocities. Using these values for the sprocket angular velocities, the vehicle rotates counter clock wise as result of opposite rotation directions of right and left sprockets, the upper part of the right side of the track chain is loose, and the upper part of the left side of the track chain is tight as shown in Figure 20. Figure 21 shows the forces of contact between side wall of the wheels and center guide of a track link.

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Figure 20. Tension adjuster force

Figure 21. Track center guide and wheel contact forces



6.9. SUMMARY AND CONCLUSIONS

The dynamics of a high speed, high mobility multibody tracked vehicle is

investigated in this paper. Compliant forces are used to define the connectivity between the links of the track chains instead of an ideal pin joint. Two track link models are considered in this study. These are the single pin and double pin track models. In the single pin track model, only one pin is used to connect two track links in the chain. In the double pin track model, two pins are used with a connector element to connect two links of the track chain. Rubber bushings are used between the track links and the pins. The stiffness and damping characteristics of the contact forces are obtained using experimental testing. By using experimental data, the generalized contact and bushing forces associated with the generalized co-ordinates of the tracked vehicle are developed. The tracked vehicle model used in this investigation includes significant details that include modeling the chassis, sprockets, idlers, road wheels, road arms, and the multi-degree of freedom track chains. The vehicle model is assumed to consist of 189 bodies, 36 pin joints, and 152 bushing elements. The model has 954 degrees of freedom. Because of the high frequency contact forces, numerical difficulties are often encountered in the simulation of multibody tracked vehicles. An explicit numerical integration method that has a large stability region is employed in this study. The method employs a variable time step size in order to achieve better computational efficiency. It was observed that the time step size significantly decreases as the vehicle speed increases. Several simulation scenarios are examined in this investigation. These include accelerated motion, high speed motion with a constant velocity, braking, and turning motion. The simulation results demonstrate the significant effect of the bushing stiffness on the dynamic response of the multibody tracked vehicle. It was also shown that the use of the double pin track model leads to a significant reduction in the bushing forces as compared to the single pin track model.

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REFERENCES

(1) Galaitsis A.G., 1984, “A Model for Predicting Dynamic Track Loads in Military

Vehicles,” ASME, Journal of Vibration,Acoustics, Stress, and Reliability in Design, Vol.

106/289

(2) Bando, K., Yoshida, K., and Hori, K., 1991, “The Development of the Rubber Track

for small Size Bulldozers,” International off-Highway Powerplants Congress and

Exposition, Milwaukee, WI, Sept. 9-12

(3) Nakanishi, T., and Shabana,(1994)"Contact Forces in The Nonlinear Dynamic Analysis

of Tracked Vehicle," International Journal For Numerical Methods in Engineering, 1994

1251-1275.

(4) Choi, J. H., 1996 "Use of Recursive and Approximation Methods in The Dynamic

Analysis of Spatial Tracked Vehicle," Ph. D. Thesis, The University of Illinois at Chicago

Scholar C. and Perkins N., 1997, “Longitudinal Vibration of Elastic Vehicle Track

System”

(5) Newmark NM. “A method of computation for structural dynamics.” Journal of the

Engineering Mechanics Division, ASCE 1959; 85 (EM3):67-94

(6) J. Chung, J. M. Lee,(1994) “A New Family of Explicit Time Integration Methods for

Linear and Non-linear Structural Dynamics,” International Journal for Numerical

Methods in Engineering, Vol.37, 3961-3976

(7) J. Chung,(1992) “Numerically Dissipative Time Integration Algorithms for Structural

Dynamics,” Ph.D. dissertation, University of Michigan, Ann Arbor

(8) Shabana A,(1989) “Dynamics of Multibody Systems,” John Wiley & Sons, New York

(9) Shabana, A,(1996) “Theory of Vibration, An Introduction,” Second Edition, Springer-

Verlag, New York

(10) Park DC, Seo IS, Choi JH. Experimental study on the contact stiffness and damping

coefficients of the high mobility multibody tracked vehicle. Journal of Korea Society of

Automotive Engineers 1999; 7:348-357

(11) Lee, H. C., Choi., J. H., Shabana, A. A., Feb. 1998 "Spatial Dynamics of Multibody



Tracked Vehicles: Contact Forces and Simulation Results," Vehicle System Dynamics,

Vol. 29, pp. 113-137

(12) E. L. Wilson,(1968) “A Computer Program for the Dynamic Stress Analysis of

Underground Structures,” SESM Report No. 68-1, Division of Structural Engineering

and Structural Mechanics, University of California, Berkeley

(13) K. C. Park and P. G. Underwood,(1980) “A Varialbe-step Centeral Difference Method

For Structural Dynamics Analysis – Part 1. Theoretical Aspects,” Computer Methods in

Applied Mechanics and Engineering 22, 241-258

(14) Owen J. Guidelines for the Design of Combat Vehicle Tracks. Dew Engineering and

Development Ltd., Ottawa, Canada.

(15) Choi, J. H., Lee., H. C., Shabana, A. A., Jan. 1998 "Spatial Dynamics of Multibody

Tracked Vehicles: Spatial Equations of Motion," International Journal of Vehicle

Mechanics and Mobility, Vol. 29, pp. 27-49

(16) Bruce Maclaurin (1983) “Progress in British Tracked Vehicle Suspension Systems,”

830442 Society of Automotive Engineers(SAE)

(17) Ketting Michael,(1997) “Structural Design of Tension Units for Tracked Vehicles,

especially Construction Machines Under The aspect of Safety Requirements,” Journal of

Terramechanics, Vol. 34, No. 3, pp. 155-163.

(18) Trusty RM, Wilt MD, Carter GW, Lesuer DR. Field measurement of tension in a T-142

tank track. Experimental techniques, 1988.

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7

DYNAMIC TRACK TENSION OF HIGH MOBILITY TRACKED VEHICLES

7.1. INTRODUCTION

The track tension of tracked vehicles plays significant roles for the dynamic behaviors, such as separation of the track system from the chassis system, distribution of wheel supporting pressure, power efficiency, vibration and noise, and the life of the track system. Due to the importance of the track tension in designing tracked vehicles, study of the dynamic track tension has long been a subject for many researchers in manufacturers and academia. However, it is very difficult to clearly understand the nonlinear behaviors of the dynamic track tension while a vehicle runs, even though both experimental and numerical works have been attempted [1-7].

Both numerical and experimental investigations are carried out in this paper. For the experimental investigation, strain gages are attached on track pin-bushing locations of track shoe body, and signal processing and recording modules are installed on the inside of track shoe body. Only limited results can be collected through the experiment due to small installation space inside of a track shoe body, high impulsive shock and vibration, and high temperature over 150 cent degrees. In order to make up the limitation of experimental results, a tracked vehicle model developed in [9] is used to obtain various numerical results. Each track link is modeled as a body which has six degrees of freedom and is connected by a bushing force element. The numerical results are validated against the experimental results before they are used for investigations.

Doyle and Workman [1] presented a static prediction of track tension when the suspensioned-tracked vehicle traverses obstacles using two dimensional finite element methods. An elastic beam element subjected to tension, compression and bending loads was utilized to model track links. Galaitsis [2]



demonstrated that the analytically predicted dynamic track tension and suspension loads of a high speed tracked vehicle are useful in evaluating the dynamic analysis of the vehicle. The predicted track tensions were compared with the empirically measured track tensions. A detailed track tension measurement methodology and results are presented by Trusty et al [3]. Strain gages connected to a portable data acquisition system were installed in the track link. The flat ground, quick acceleration, traversal of obstacle courses, pivot turns, moving uphill, and pre and post tension, were used for the tension investigation scenarios. McCullough and Haug [4] designed a super element that represents spatial dynamics of high mobility tracked vehicle suspension systems. The track was modeled as an internal force element that acts between ground, wheels and the chassis of the vehicle. Track tension was computed from a relaxed catenary relationship. Empirical normal and shear force formulas based on constitutive relations from soil mechanics were used to model the soil-track interface. Choi [5, 6] presented a large scale multibody dynamic model of a construction tracked vehicle in which the track is assumed to consist of track links connected by single degree of freedom pin joints. In this detailed three dimensional dynamic model, each track link, sprocket, roller, and idler is considered as a rigid body that has a relative rotational degree of freedom. Scholar and Perkins [7] developed an efficient alternative model of the track chains considering longitudinal vibrations. The track is assumed to consist of a finite number of segments, each of which is modeled as a continuous uniform elastic rod attached to the vehicle wheels. Overall chain stretching effects are accounted for.

The purpose of this paper is to investigate the dynamic track tensions of a high mobility tracked vehicle maneuvering under various driving conditions. Both numerical and empirical methods are employed and the effects of pretensions, friction forces, interacting proving grounds, vehicle speeds, and driving torque are explored for the sake of understanding dynamic behaviors of the track system.

7-3

7.2. NUMERICAL MODEL OF A HIGH MOBILITY TRACKED VEHICLE

Turret Chassis system

Track system

Figure 1. High mobility multibody tracked vehicle model

A three-dimensional multibody tracked vehicle model shown in Fig. 1 consists

of a chassis subsystem and two track subsystems. The chassis subsystem includes a chassis, sprockets, support rollers, idlers, road arms, road wheels and the suspension units. The sprockets, support rollers, and road arms are connected to the chassis by revolute joints. The track link subsystem includes a shoe body, a pin, rubber bushings, and a rubber pad. Rubber bushings and pin are inserted into the hole of a shoe body with a radial pre-pressure and a rubber pad is mounted on the ground interaction side of the shoe body. The vehicle model used in this investigation consists of 189 bodies; 37 bodies for the chassis subsystem, 76 bodies for each track subsystem, 36 revolute joints and 152 bushing elements and has 954 degrees of freedom.



Figure 2. Hydraulic track tension adjustor system

Suspension systems and tension adjustor: The suspension units of the

vehicle include a Hydro-pneumatic Suspension Unit(HSU), and torsion bar that are modeled as force elements whose compliance characteristics are obtained from analytical and empirical methods. The HSU systems are mounted on first, second, and sixth stations to damp out pitching motion and to decrease an impact when the vehicle is running over large obstacles. The torsion bars are mounted on the middle stations for this vehicle model. A simple torsional spring model is used in this investigation to represent the stiffness of the torsional bars. The hydraulic passive tension adjustor is installed on the idler to maintain a proper track tension of the tracked vehicle model. Figure 2 shows the schematic diagram of the tensioner system of the vehicle. The hydraulic ram of the tension adjustor is modeled as an equivalent linear spring-damper force element.

Track link connection: Each track subsystem is modeled as a series of bodies connected by rubber bushings around the link pins which are inserted into a shoe plate with a radial pressure to reduce rattling of the pin. When the vehicle runs over rough surfaces, the track systems are subjected to extremely high impulsive contact forces as the result of their interaction with the vehicle components such as road wheels, idlers, and sprocket teeth, as well as the ground. The rubber bushings tend to reduce the high impulsive contact forces by providing cushion and reducing the relative angle between the track links. In this

7-5

investigation, a continuous force model is used to represent the pin connections. This force model is a non-linear function of the coordinates of the two links. Note that because of the bushing effect, the origins of the joint and pin coordinate systems do not always coincide.

Contact detection and forces: In this section, the contact force model and the contact detection algorithms between the track links and the road wheels, rollers, and sprockets are briefly discussed. A more detailed discussions on the formulation of the contact force model is presented by Choi, et al, [5, 6, 9]. As shown in Fig. 1, when a track link travels around vehicle components, its trajectory is determined by the contact forces. These forces are created by detecting on contact conditions. The contact detection algorithms monitor the contacts of, wheel and track link contact, center guide and wheel contact, sprocket tooth and track link pin contact, and side wall of track link and sprocket contact. Once a contact condition is satisfied, contact forces are applied at the contacted position to restitute each other.

Figure 3. Interaction between track shoe body and triangular patch element



Interacting ground representations: The ground interacting surface of a track

link can be single or multiple, and therefore, there are one surface or multiple surfaces on each track link that can come into contact with the ground. The interacting surface of ground is discretized and each contact node points were defined. The global position vectors that define the locations of points on the shoe plates surface of track link are expressed in terms of the generalized coordinates of the track links and are used to predict whether or not the track link is in contact with the ground. Since the contact surface of track link consists of rubber pad and steel shoe plate, the contact forces at each node point are evaluated by using their own stiffness and damping coefficients. In order to construct various geometries of tracked vehicle paved proving ground [10], such as bumping courses, trench course, inclined course, standard cross country courses, descritized terrain representation methods using triangular patch element are used in this investigation. The plane equation of interacting ground profiles for a triangular patch element which has three nodes and a unit normal vector is employed as illustrated in Fig. 3.

Equations of motion: Since the track system interacts with the chassis components through the contact forces and adjacent track links are connected by compliant force elements, each track link in the track system has six degrees of freedom which are represented by three translational coordinates and three Euler angles [11]. The equations of motion of the chassis that employs the velocity transformation defined by Choi [5, 9] are given as follows:

)( rii qBMQBqMBB TT &&&& −= (1)

where , B and q are relative independent coordinates, velocity transformation matrix, and Cartesian velocities of the chassis subsystem, and M is the mass matrix, and Q is the generalized external and internal force vector of the chassis subsystem, respectively. Since there is no kinematic coupling between the chassis subsystem and track subsystem the equations of motion of the track subsystem can be written simply as

riq &

7-7

ttt Qq =&&M (2)

where , and denote the mass matrix; and the generalized coordinate and force vectors for the track subsystem. Consequently, the accelerations of the chassis and the track links can obtained by solving Eqs. 1 and 2.

tM tq tQ

G-Alpha integrator : Many different types of integration methods can be employed for solving the equations of motion for mechanical systems. Explicit methods have small stability region and are often suitable for smooth systems whose magnitude of eigenvalues is relatively small. Contrast to the explicit methods, implicit methods have large stability region and are suitable for stiff systems whose magnitude of eigenvalues is large. One of the important features of the implicit methods is the numerical dissipation. Responses of mechanical systems beyond a certain frequency may not be real, but be artificially introduced during modeling process. In the model used in this investigation, a contact between two bodies is modeled by compliance elements. Lumped characteristics of the spring and damper must represent elastic and plastic deformations, and hysterisis of a material. Such characteristics may include artificial high frequencies which are not concern of a design engineer. Unless such artificial high frequency is filtered, an integration stepsize must be reduced so small that integration can not be completed in a practical design cycle of a mechanical system. To achieve this goal, generalized-alpha method [8, 9] has developed to filter frequencies beyond a certain level and to dissipate an undesirable excitation of a response. One of the nice advantages of the generalized-alpha method is that the filtering frequency and dissipation amount can be freely controlled by varying a parameter in the integration formula. As a result, the generalized-alpha method is the most suitable integration method for integrating the equations of motion for stiff mechanical systems. Figure 4 shows the animation of high mobility tracked vehicle when the vehicle runs over a trench profile.



(a) time = 0.0 sec (b) time = 3.0 sec

(c) time = 6.0 sec (d) time = 9.0 sec Figure 4. Computer animation of multibody tracked vehicle running

over trench ground profile 7.3. INTERACTION GROUNDS

The ground interacting surface of a track chain link can be single or multiple,

and therefore, there are one surface or multiple surfaces on each track link that can come into contact with the ground. The interacting surface of chain link is discretized and each contact node points were defined. The global position vectors that define the locations of points on the shoe plates surface of chain link are expressed in terms of the generalized coordinates of the track chain links and are used to predict whether or not the track chain link is in contact with the ground. Since the contact surface of track chain link consists of rubber pad and steel shoe plate, the contact forces at each node point are evaluated by using their own stiffness and damping coefficients. In order to construct various geometries of tracked vehicle paved proving ground, such as bumping courses, trench course, inclined course, standard cross country courses, discretized terrain representation methods using triangular patch element are used in this investigation. A triangular patch element has three nodes and a unit normal vector to describe plane equations of interaction grounds [13].

7-9

(a) Series of triangular patch for generalized virtual body

(b) Triangular patch surface

Figure 5. Discretized terrain representation

Discretized terrain representation: The virtual terrain model used in this

investigation is a general three dimensional surface defined as a series of triangular patch elements. Figure 5 (a) shows an example of virtual ground using 8 points and 6 elements. Most geometries of various paved proving ground for tracked vehicle can be represented by using triangular patch elements. The equation for the plane defined from three nodes can be written as

zxaxaxa =++ 321 (3)

The three coefficients, a , , and of the equations of plane can be obtained by given three locations of triangular patch shown in Fig. 5 (b), and by using Cramer's rule [14], these coefficients are

1 2a 3a



AA

detdet k

ka = , = 1, 2, 3 (4) k

where

====

×

×

×

×

333

332

331

33

][][][][

kkk

kk

kk

kk

zyxAzxAyzAyxA

III

and (5) T]111[=I

Then the unit normal vector of the plane is defined as n

Taaaa

]1[1

1ˆ 2122

21

−++

±=n (6)

Virtual proving ground : Until recent development of computer simulation model [6, 7, 9], the development process of tracked vehicle have been depended on inefficient technologies of repeated procedures ; construction prototype vehicle based on basic calculations and simple computer simulation, test on proving ground, then modification. This expensive design procedure can be diminished by recent developments of computer simulation.

In this investigation, only paved ground models are developed for the virtual test of dynamic analysis of three dimensional tracked vehicle. The developed computer models of grounds are stored into the created ground library.

As shown in Figs. 6 and 7 the variety of virtual proving grounds, symmetric and unsymmetric bump courses, trench and ditch courses, longitudinal and laternal inclined courses, and standard cross-country courses of RRC9 and Profile IV, are constructed by using triangular patch elements. When a vehicle runs over these virtually created proving grounds, the nonlinear behaviors of track chains resulting from the interacting with the test grounds are obtained in this numerical investigation.

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(a) Single bump course (d) Obstacle course

(b) Trench course (e) Grade ability slope

(c) Ditch course (f) Side slope

Figure 6. Various paved virtual proving ground using triangular patch



(a) Series of triangular bump

(b) Series of trapezoidal bump

Figure 7. Simulated cross-country course (APG Profile IV)

Methods of finite track chain-ground contact point: Unlike wheel and surface contact, the interactions between track chain link and ground are very complicated problems. This is because the track chain link has irregular contact geometry and different material properties. According to large number of track

7-13

chain links of each track subsystem, commonly used contact theory of surface to surface interactions in finite element community can not be employed for this work. Choi [15] suggested that element free finite contact nodes were distributed on the contact surface of track chain link, which have their own stiffness and damping characteristics. The relative indentations of each nodes were monitored and positions are restored. The use of element free finite contact node methods demonstrated clearly the computational efficiency for dynamic analysis of track chain system. Based on the method developed by Choi [15], the interactions between track chain link surface and triangular patch surface are developed in this paper. Figure 3 shows the interaction between finite contact nodes of track link and triangular patch surface. The perpendicular deformation scalar of

contact node

ijkd

j of link i on patch plane can be defined as k

ijk

Pijkd nr ˆ1 ⋅= (7)

where is shown in Fig. 3 and unit vector is defined in Eq. 6.. The

criterion of necessary condition for the contact to occur of node

P1r ijkn

j , which is not sufficient, is

>≤

separatedcontactd

ijk

ijk

00

(8)

If this conditions is satisfied, the position vector shown in Fig. 3 is used

to compute the node location whether contact point

jkBr

B of node j is on the

patch plane . The position vector can be written as k jkBr

ijk

ijk

ijp

iijkB duARr n−+= (9)

where is the transformation matrix associated with the orientation coordinates of link and

iAi ij

pu is the local position vector of node j in the

track chain link coordinate system. On the other hand, using scalar triple product



if one of the following conditions is satisfied

≥⋅×≥⋅×≥⋅×

000

331

223

112

nB

nB

nB

urrurrurr

or (10)

≤⋅×≤⋅×≤⋅×

000

331

223

112

nB

nB

nB

urrurrurr

then the node j of link is in contact with patch element . i k

If the node j is in contact with patch plane , the contact force at the contact node can be computed using the equation as

k

ijk

ijk

ijk

ijk

ijk dCdKF &+= (11)

where and C are, respectively, the stiffness and damping coefficients of

the contact force model at node

ijkK ij

k

j of body on patch plane . Using the expression for the contact force as defined by the preceding equation, the contact force vector can be defined as

i k

ijk

ijk

ijk F nF ˆ= (12)

where is a unit normal vector shown in Fig. 3. The virtual work of the contact force at the nodes is given by

ijkn

∑=

=n

j

ijk

ik WW

1δδ ∑

=

=n

j

ijk

ijk

ijk dF

1

ˆ δn [ ]

= i

iTiTi

R θR

QQδδ

θ (13)

where

∑=

=n

j

ijk

TiR

1FQ

∑=

=n

j

ijk

Tijp

iTi

1)~( FuAQθ (14)

are the generalized contact forces associated with the Cartesian and orientation

7-15

coordinates of link , and i ij

p is the skew symmetric matrix associated with the

vector ijpu . In order to evaluate the tangential component of these contact forces

for friction effect at each contact nodes, the smooth Coulomb friction model [6] is employed in this investigation. Figure 8 shows the computer animation of multibody tracked vehicle running over APG Profile IV test ground.

u~

Note that the proposed element free finite contact node method have several advantages such as, simple computer implementation, easy contact detecting algorithm for irregular surface, independent contact coefficients, and distribution of concentrated contact forces, however, in the penalty function approach used in this contact force model the determinations of spring and damping coefficients may be a black art. These coefficients may not correspond to familiar physical properties that can be measured experimentally. Careful numerical calibration process is necessary to obtain reliable model, accordingly.

Time = 0.0 sec Time = 9.0sec

Time = 3.0 sec Time = 12.0 sec



Time = 6.0 sec Time = 15.0 sec

Figure 8. Computer animation of multibody tracked vehicle running over Aberdeen

profile IV proving ground

7.4. MEASUREMENT OF THE DYNAMIC TRACK

The measurement system is composed of strain gages, signal processor, data storage, and power unit. The system is installed inside of a track shoe body. When the switch is on, the system will start to measure and store the tensional forces from the strain gages into data storage processor. The measurement results are then downloaded into a laptop computer through communication port.

The basic platform of dynamic track instrumentation system is developed by Kweenaw Research Center at Michigan Technology University [12]. The tension measurement system records 2 channels, which are track tensions at both ends of a track link, at the rate of 800 samples per second for 160 seconds. The tension data in the system memory is offloaded to a computer for storage after the test vehicle is stopped. A track shoe body was carved to attach full bridge of strain gages on the outside and inside edges of the body. A track link, as a sensor system to measure the dynamic track tension of the high mobility tracked vehicle, was carefully calibrated at the center. A known load, Shunted Engineering Unit Value, can be simulated by shunting one leg of the strain gage bridge using a 58,900 ohm resistor inside the measurement system. The known load is about 20,000 lb [12]. If any load is applied to the measurement system, the load as an engineering unit can be determined by a linear interpolation or extrapolation using the engineering unit value.

Figure 9 shows the comparison of simulation and experimental results when the vehicle runs on flat ground with the velocity of 10 km/h. The figure shows

7-17

that there are four disagreement areas between experimental and numerical results. These disagreements are due to the extra deformation of the strain gage when the track link moves around sprocket, idler, and first and last road wheels. The extra deformation makes the track tension look much higher than it actually is

12 13 14 15 16 170

10k

20k

30k

40k

50k

60k

70k

80k

Time(sec)

Tens

ion(

N)

Experiment

Simulation

Figure 9. Dynamic tension of a track link

7.5. NUMERICAL INVESTIGATION OF DYNAMIC TRACK TENSION

Extended numerical simulations are carried out to compensate for the

experimental limitations due to space and environment. The track tension is monitored in two different views of track link following view and chassis fixed view. In order to acquire the track tension for the chassis fixed view, the track tensions are recorded until all links pass through one point of the hull. For the track link following view, the track tension of one selected track link is recorded when it is moving around vehicle components of idler, road wheels, sprocket and support rollers.

Key physical quantities influencing the track tension are pre-tension, vehicle speed, ground profile, traction force, driving torque, and turning resistance, respectively. The pre-tensions of 25 kN, 50 kN and 100 kN are given to observe their influences on the dynamics of the vehicle. Three different speeds of 5 km/h, 20 km/h, and 40 km/h are given on both driving sprockets using velocity



constraint equations. Various ground profiles as defined in real proving ground [10] are developed by using the triangular patch elements. Three friction coefficients of 0.1, 0.4, and 0.7, between the track shoe body and the ground, are used for different traction force modeling. The track tensions are observed for a pivot turning, right and left turning, backward motion, acceleration and braking motions.

Effect of pre-tension: One of the most critical variables for the dynamic track tension is the pre-tension. Although an optimal pre-tension has long been a major subject for academia and industry, researchers only relied on experimental and field experiences. Most of high mobility suspenioned tracked vehicles, approximately 10 % of the vehicle total weight is loaded as a track pre-tension. Figure 10 shows the track tensions of a selected track link in the link following view with three different pre-tensions. These pre-tensions are 25 kN, 50 kN, and 100 kN, respectively. Both sprockets have constant angular velocity of -17.8 rad/sec which can produce 20 km/h vehicle speed. As illustrated in this figure, increment of the pre-tension linearly increases the dynamic track tension.

0 1 2 3 40

20k

40k

60k

80k

100k

120k

5

50kN pre-tension

100kN pre-tension

25kN pre-tension

Ten

sion

(N)

Time(sec)

Figure 10. Track tensions of pre-tension effect

7-19

Effect of vehicle speed: Like a tire of wheeled vehicles, revolution of a track system can cause the movement of tracked vehicles. The vehicle speed varies time to time due to random and irregular vehicle operations. Several numerical and empirical studies showed that the amplitude of track tension does not change much as the speed changes. However, the frequency of the track tension changes significantly. In the case of a bump run, the track tension around contacted region increases significantly at a higher speed when the vehicle hits a bump. It is mainly because of large increment of impact force between a track link and the ground.

Effect of ground profile: In the previous section, a generalized method for building the proving ground profiles are introduced. Many profiles representing real testing grounds are developed by using the triangular patch. Since the ground contact forces are directly transferred to the track links, the track tension is strongly related to the surface geometry of a given ground profile.

Effect of traction forces: Force transmission of a tracked vehicle can be understood as three force conversions. When a track system rotates, traction forces are generated between the track system and the ground in opposite direction to the velocity of the track system. Contact forces between track link pins and sprocket teeth are converted to the sprocket moment in the tangential direction of the pitch circle of the sprocket. The sprocket torque can be converted again to a translational force acting on the axis of the sprocket center. Finally, this translational force on the axis of the sprocket can cause movement of a tracked vehicle. During the force conversion process the traction forces can be replaced directly by tensional forces of track system. The amount of the traction forces is determined by a friction model between the track system and the ground. The track tensions between middle road wheels with different friction coefficients are shown in Fig. 11. Three different friction coefficients, 0.1, 0.4, and 0.7 are used in this numerical investigation. In order to show the effect of the friction, the vehicle is accelerated from zero to 40 km/h in ten seconds. As shown in this figure, increment of the friction coefficient causes an increment of the track tension.



0.0 0.5 1.0 1.5 2.0 2.5 3.05x104

6x104

7x104

8x104

µ=0.4

µ=0.7

µ=0.1

Ten

sion

(N)

Time(sec)

Figure 11. Track tensions of traction force effect

Effect of sprocket torque : In this investigation power-pack, engine and transmission, are modeled by using velocity constraints or the sprocket torque. In the real world there are two major disturbances to keep steady sprocket torque. These are irregular driver inputs and impacts of transmission shifts. The sprocket torque is converted to the contact force between the sprocket teeth and track link pins. The sprocket contact force repeats to pull and push, which makes the track tension vary. To observe the effects of sprocket torque, step, sinusoidal, and linear-steady torques are applied on the sprocket. Figure 12 shows the track tension changes near the sprocket when the step torque is applied.

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1 2 3 4 5 6 7 8 9 10 110

5k

10k

15k

20k

Tor

que(

Nm

)

Time(sec)

20k

30k

40k

50k

60k

tension

torque

Tension(N

)

Figure 12. Track tensions of sprocket torque effect

0 2 4 6 8 10 12

0

50k

100k

150k

200k

250k

road wheels

upper

right track tension

left track tension

sprocket

upper

Ten

sion

(N)

Time (sec)

Figure 13. Track tensions of turning resistance effect

Effect of turning motion: Heading direction of a tracked vehicle is turned by a speed difference of left and right sprockets, which causes different traction forces. The traction forces of track systems are converted to the forces at the center of both sprocket axes. Then the chassis system is rotated with respect to the vertical axis by a force difference of both sprocket axes. Suppose a track vehicle is stuck to the ground. If angular velocities of both sprockets are constant



with different speeds, significant differences of track tensions may be observed due to the revolution of the chassis.

Figure 13 shows the dynamic changes of right and left track tensions when the vehicle makes right turning with angular velocities of = -4.5 rad/sec and

= -5.4 rad/sec. It can be shown that the track tension of the upper part of

the left sprocket goes up significantly, while the track tension of the lower part goes down.

rightω

leftω

7.6. FUTURE WORK AND CONCLUSIONS

The dynamic track tension for a high mobility tracked vehicle is investigated in this paper. The three dimensional multibody tracked vehicle consists of the hull, sprockets, road arms, road wheels, support rollers, and sophisticated suspension systems of hydro-pneumatic and torsion bars. A compliant force model is used to connect the rigid body track links. The tracked vehicle model has 189 bodies, 36 pin joints and 152 compliant bushing elements and has 954 degrees of freedom. Various ground profiles are developed by using triangular patch elements. Numerical results are validated against experimental results. Numerical simulations have been carried out under various maneuvering conditions and effects of several conditions are discussed . Numerical results showed that the optimal track tension may not be necessarily 10 % of the total vehicle weight as many track vehicle researchers have believed. Further studies must be carried out to find the optimal track tension.

7-23

REFERENCES

[1] G. R. Doyle and G. H. Workman, 1979, ''Prediction of Track Tension when Traversing

an stacle'', Society of Automotive Engineers, 790416

[2] A.G. Galaitsis, 1984, "A Model for Predicting Dynamic Track Loads in Military

Vehicles", ASME, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol.

106/289

[3] R.M. Trusty, M.D. Wilt, G.W. Carter, D.R. Lesuer, 1988, ''Field Measurement of Tension

in a T-142 Tank Track'', Experimental Techniques

[4] M. K. McCullough, and E. J. Haug, 1986, ''Dynamics of High Mobility Tracked

Vehicles'', ASME, Journal of Mechanisms Transmissions, and Automation in Design,

Vol.108, pp. 189-196

[5] Choi, J. H., Lee., H. C., Shabana, A. A., Jan. 1998, ''Spatial Dynamics of Multibody

Tracked Vehicles: Spatial Equations of Motion'', International Journal of Vehicle

Mechanics and Mobility, Vehicle System Dynamics, Vol. 29, pp. 27-49

[6] Lee, H. C., Choi., J. H., Shabana, A. A., Feb. 1998, ''Spatial Dynamics of Multibody

Tracked Vehicles: Contact Forces and Simulation Results'', International Journal of

Vehicle Mechanics and Mobility, Vehicle System Dynamics, Vol. 29, pp. 113-137

[7] C. Scholar and N. Perkins, 1997, "Longitudinal Vibration of Elastic Vehicle Track

System", SAE, 971090, International Congress and Exposition, Detroit, MI, Feb. 24-27

[8] J. Chung, J. M. Lee, 1994 ''A New Family of Explicit Time Integration Methods for

Linear and Non-linear Structural Dynamics'', International Journal for Numerical

Methods in Engineering, Vol.37, 3961-3976

[9] H. S. Ryu, D. S. Bae, J. H. Choi and A. Shabana, 2000 ''A Compliant Track Model For

High Speed, High Mobility Tracked Vehicle'', International Journal For Numerical

Methods in Engineering, Vol. 48, 1481-1502

[10] Changwon Proving Ground Construction Manual, 1996, Agency for Defense

Development, GWSD-809-960634

[11] Shabana A, 1989 ''Dynamics of Multibody Systems'', John Wiley & Sons, New York



[12] Glen Simula, Nils Ruonavaara, and Jim Pakkals, 1999 "DTIS operation manual",

KRC, Michigan Technological University.

[13] ADAMS Reference Manual, Mechanical Dynamics, 2301 Commonwealth Blvd, Ann

Arbor, MI 48105.

[14] E. Kreyszig, 1983 “Advanced Engineering Mathematics”, 5th edition John Wiley &

Sons, New York

[15] Choi, J. H., 1996 “Use of Recursive and Approximation Method in the Dynamic

Analysis of Spatial Tracked Vehicle”, Ph. D. Thesis, The University of Illinois at

Chicago

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8

EFFICIENT CONTACT AND NONLINEAR DYNAMIC MODELING FOR TRACKED VEHICLES

8.1. INTRODUCTION

In the dynamic analysis of vehicle system the mathematical modeling for the system can be very different according to the objective of analysis. Sometimes the mathematical modeling methods of vehicle systems are pursuing more simplified effective models such as for real-time analysis or for the system design which doesn’t require highly nonlinear effects. Oppositely, due to rapid developments of computer hardware and numerical technologies, researchers and engineers can construct super detail nonlinear dynamic models which have several hundreds, even several thousands of degrees of freedom systems, which has the same phenomena as physical system. The objective of this research is to built a reliable tracked vehicle dynamic analysis model so as to design a dynamic track tensioning control system for high speed tracked vehicle based on multibody dynamic modeling techniques. One of the key points for the dynamic analysis of tracked vehicle is to predict the dynamic track tension when the vehicle operates on various ground. In order to satisfy such objective of the research, the track links of the track system should be modeled as a rigid body which has six degrees of freedom connected by bushing force elements.

In early 80’s several dynamic modeling techniques for track systems have been developed in universities, research institutes and companies. McCullough and Haug[1] designed a super element that represents spatial dynamics of high mobility tracked vehicle suspension systems. The track was modeled as an internal force element that acts between ground, wheels and the chassis of the vehicle. Track tension was computed from a relaxed catenary relationship. Empirical normal and shear force formulas based on constitutive relations from soil mechanics were used to model the soil-track interface. Frank Huck[2]



introduced a planar multibody dynamic model of track type tractor by using DRAM software. In this investigation each track link was modeled as a rigid body and contact force analyses of sprocket, rollers and soil ground were represented as a pioneer works. Similar modeling technique was also developed by Tajima[3] at the similar period. The Komatsu Ltd. in-house multibody program developed by Tajima is used to simulate planar multibody tracked vehicles. The contact search mechanics and dynamic analysis of planar rigid body track system are clearly introduced by Nakanishi and Shabana[4]. Nakanishi’s work was extended for three dimensional analysis by Choi and Shabana[5, 6], and simultaneously Wehage[7] also developed the full three dimensional tracked vehicle model, which considers the track link as a rigid body, under research project in Caterpillar Inc. Whereas Choi used to connect each rigid tracked link by one degree of freedom pin joint, bushing force elements were used to connect for rigid track link by Wehage. Choi’s work shows an possibility of very difficult numerical solution however it fails to give more freedoms in real world than Wehage’s approach. Ryu et al.[8] extended previously developed track system modeling techniques for the high mobility military tracked vehicle which adopts sophisticated suspension and tensioning systems. In this investigation a new variable step algorithm is implemented into G-Alpha integrator which gives high numerical damping to integrate smoothly high frequency and impulsive contact and bushing forces.

There are several reasons why many researchers has tried to develop rigid multibody track systems even though such modeling techniques burden heavily for numerical solutions. Unlike tires of wheeled vehicles track system causes many problems such as separations or failures of connections, etc., furthermore it is very expensive to maintain and has relatively weak durability. Because of superiority of track system on very hostile terrain it cannot be replaced by wheeled system, thus researchers should have solved these difficulties of the tracked vehicle system. In the beginning of the research several simple modeling techniques had been introduced however those gave a conclusion that each track link should be considered as a rigid body to satisfy requirements. For instant, one of the key issue for tracked vehicle is track tension since track tension has significant roles for the vehicle maneuverings as focused in this investigation. Very few works have been performed for the analysis of track tension based on

8-3

empirical or simple numerical analysis. Doyle and Workman[9] presented a static prediction of track tension when the suspensioned tracked vehicle traverses obstacles using two dimensional finite element methods. An elastic beam element subjected to tension, compression and bending loads was utilized to model track links. Galaitsis[10] demonstrated that the analytically predicted dynamic track tension and suspension loads of a high-speed tracked vehicle are useful in evaluating the dynamic analysis of the vehicle. The predicted track tensions were compared with the empirically measured track tensions. A detailed track tension measurement methodology and results are presented by Trusty et al.[11]. Strain gages connected to a portable data acquisition system were installed in the track link. The flat ground, quick acceleration, traversal of obstacle courses, pivot turns, moving uphill, and pre and post tension, were used for the tension investigation scenarios. Choi et al.[12] predicted and showed the effect of dynamic track tension for the vehicle by using multibody techniques. This research focuses on a heavy military tracked vehicle which has sophisticated suspension and rubber bushed track systems. Various virtual proving ground models are developed to observe dynamic changes of the track tension. The predicted dynamic track tensions are validated against the experimental measurements.

In this investigation for the sake of efficient development of dynamic track tensioning system for suspensioned high speed military tracked vehicle, detail nonlinear dynamic modeling methods which can partially replace physical prototype models are presented. For the multibody dynamic modeling techniques of the tracked vehicle used in this research several new methods are developed and suggested. Those are efficient contact detecting kinematics for sprockets, wheels and track links, parameter extraction techniques from component experimental test, and a method how to apply Bekker’s[13] soil theory for multibody track and soil interactions. The simulation results are correlated by newly developed experimental measurement techniques in this investigation.



8.2. MULTIBODY TRACKED VEHICLE MODEL AND PARAMETER

EXTRACTIONS

The tracked vehicle model used in this investigation is a military purpose high speed tank system which has sophisticated suspension system to damp out impacts from hostile ground. In general this type of vehicle can be divided four subsystems for overall motion analysis of vehicle dynamics. These subsystems are two track subsystem with suspension units, main body subsystem with power pack, and turret subsystem with main gun. The each right and left track subsystems is composed of rubber bushed track link, double sprockets with single retainer, seven road wheels and arms, and three upper rollers. The sprockets, road arms, road wheels, upper rollers and turrets are mounted on main body by revolute joints which allow single degrees of freedom. Total 38 revolute joints are used for the vehicle modeling and generate 190 nonlinear algebraic constraint equations. Two busing force elements to connect each track links and total 304 bushing forces elements for both track systems are used in this investigation. The modeled vehicle has 191 rigid bodies and 956 degrees of freedom. Figure 1 shows a computer graphics model for tracked vehicle used in this investigation.

Turret system

Chassis system

Track system

Figure 1. Computer graphics of high speed tracked vehicle model

8-5

8.3. EFFICIENT CONTACT SEARCH ALGORITHM

The interactions between the track links and the road wheels, rollers, and

sprockets are explained in this section. When a track link travels around vehicle components, its trajectory is controlled by contact forces. The contact forces can be generated computationally by detecting of contact conditions. The contact collision algorithms are composed of five main routines such as search routines for, wheel and link contact, center guide and wheel contact, sprocket tooth and link pin contact, side wall of link and sprocket contact, and ground and link shoe. The contact points and penetration values are defined from the searching routines. Then a concentrated contact force is used at the contacted position of the contact surface of the bodies. A detailed discussion on the formulation of the contact collision is represented by Choi et al[5,6] and Nakanishi and Shabana[4]. However, it is not efficient for each chassis component such as road wheels and sprockets to search all track links in detail. Efficient search algorithms and discretized terrain representation method are investigated, respectively. 8.3.1 ROAD WHEEL-TRACK LINK CONTACT

Each road wheel is usually composed of two wheels. The interactions between road wheel and track link can be divided into two types contact, as shown in Fig. 4. One is road wheel-track link body contact and the other is wheel side-track link center guide contact. Each track subsystem has 6 road wheels and 76 track links. In order to search wheel-track link contact efficiently, the pre-search and post-search algorithm is applied. In the pre-search, bounding circle relative to road wheel center is defined. All of track links are considered to detect a starting link and ending link which has a possibility of wheel contact. Post-search means a detailed contact inspection for track links in a bounding circle. Once a starting and an ending link are found at one time through pre-search prior to analysis, only detailed search is carried out by using the information of starting link and ending link from the next time step.



Search direction

Bounding circle

ending link

starting link

Search direction

Figure 4. Wheel and track link interactions

8.3.2 SPROCKET-TRACK LINK CONTACT The interactions between sprocket and track link can be divided into two types

contact, as shown in Fig. 5. One is sprocket-track link pin(end connector) contact and the other is sprocket side-track link side. Each track subsystem has 1 sprocket and 76 track links, moreover a sprocket has many teeth. For the efficient search of the sprocket-track link contact, contact search algorithm is composed of the pre-search and post-search. In the pre-search, bounding circle relative to sprocket center is defined. All of track links are employed to detect a starting link and ending link which has a possibility of sprocket contact. Then, track links from starting link are investigated the engagement with sprocket valley. Post-search means a detailed contact inspection for track links in a bounding circle. Once a starting and ending link is found at one time through pre-search prior to analysis, only detailed search is carried out by using the information of starting link and ending link from the next time step.

8-7

ending engagement

Search direction

Bounding circle

sSearch direction

tarting engagement

ending link

starting link

Figure 5. Sprocket and track link interactions

8.3.3 GROUND-TRACK LINK CONTACT

The ground interacting surface of a track link can be single or multiple, and therefore, there are one surface or multiple surfaces on each track link that can come into contact with the ground. The interacting surface of track link is discretized and each contact node points were defined. The global position vectors that define the locations of points on the shoe plates surface of track link are expressed in terms of the generalized coordinates of the track links and are used to predict whether or not the track chain link is in contact with the ground. In order to construct various geometries of tracked vehicle paved proving ground, such as bumping courses, trench course, inclined course, standard cross country courses, discretized terrain representation methods using triangular patch element are used in this investigation. A triangular patch element has three nodes and a unit normal vector to describe plane equations of interaction grounds [16].



DISCRETIZED TERRAIN REPRESENTATION The virtual terrain model used in this investigation is a general three dimensional surface defined as a series of triangular patch elements. Figure 6 shows an example of obstacle course created by triangular patch surfaces. Most geometries of various paved proving grounds for tracked vehicle can be easily represented by using triangular patch.

Figure 6. Terrain representation (obstacle course)

The equation for the plane defined from three nodes can be written as

zayaxa =++ 321 (3)

The three coefficients, , , and of the equations of plane can be obtained by given three locations of triangular patch, and by using Cramer's rule, these coefficients can be obtained[16].

1a 2a 3a

Then the unit normal vector of the plane is defined as n

[ Taaaa

11

1ˆ 2122

21

−++

±=n ] (4)

METHODS OF FINITE CONTACT NODES FOR GROUND INTERACTIONS Unlike wheel and surface contact, the interactions between track link and

ground are very complicated problems. This is because the track link has

8-9

irregular contact geometry and different material properties. Due to large number of track links of each track subsystem, commonly used contact theory of surface to surface interactions in finite element community can not be employed for this work. Choi[17] suggested that element free finite contact nodes were distributed on the contact surface of track link, which have their own stiffness and damping characteristics. The relative indentations of each node were monitored and positions are restored. The use of element free finite contact node methods demonstrated clearly the computational efficiency for dynamic analysis of track system. Based on the method developed by Choi[17], the interactions between track link surface and triangular patch surface are developed in this investigation.

Figure 7. Interaction between track shoe body and triangular patch element

Figure 7 shows the interaction between finite contact nodes of track link and triangular patch surface. The perpendicular deformation scalar of contact node of link on patch plane can be defined as

kd

j i k



kPij

kd nr ˆ1 ⋅= (5)

where is shown in Fig. 7 and unit vector is defined in Eq. (4). The criterion of necessary condition for the contact to occur of node , which is not sufficient, is

P1r kn

j

>≤

seperatedcontactd

ijk

ijk

:0:0

(6)

If this conditions is satisfied, the position vector shown in Fig. 7 is used

to compute the node location whether contact point

jkBr

B of node is on the

patch plane . The position vector can be written as j

k jkBr

kkijp

iijkB d nuARr ˆ−+= (7)

where is the transformation matrix associated with the orientation

coordinates of link and iA

i ijpu is the local position vector of node in the

track link coordinate system. On the other hand, using scalar triple product if one of the following conditions is satisfied

j

(8)

≤⋅×≤⋅×≤⋅×

≥⋅×≥⋅×≥⋅×

0ˆ0ˆ0ˆ

0ˆ0ˆ0ˆ

331

223

112

331

223

112

kB

kB

kB

kB

kB

kB

ornrrnrrnrr

nrrnrrnrr

, then the node of link is in contact with patch element . j i k


In this investigation, the relative generalized coordinates are employed in

order to reduce the number of equations of motion and to avoid the difficulty

8-11

associated with the solution of differential and algebraic equations. Since the track chains interact with the chassis components through contact forces and adjacent track links are connected by compliant force elements, each track chain link in the track chain has six degrees of freedom which are represented by three translational coordinates and three Euler angles. Recursive kinematic equations of tracked vehicles were presented by [8] and the equations of motion of the chassis are given as follows :

)( r

iTr

iT qBMQBqMBB &&&& −= (9)

where and are relative independent coordinates, velocity

transformation matrix, and is the mass matrix, and is the generalized external and internal force vector of the chassis subsystem, respectively. Since there is no kinematic coupling between the chassis subsystem and track subsystem, the equations of motion of the track subsystem can be written simply as

riq B

M Q

ttt QqM =&& (10)

where , and Q denote the mass matrix, the generalized coordinate and force vectors for the track subsystem, respectively. Consequently, the accelerations of the chassis and the track links can be obtained by solving Eqs. (9) and (10)..

tM tq t

8.5. EXTENDED BEKKER’S SOIL MODEL FOR MULTIBODY TRACK

SYSTEM

The interactions between track link and soil used in this investigation consist of the normal pressure-sinkage and shear stress-shear displacement relationships. Bekker[13] developed the bevameter technique to measure terrain characteristics by the plate penetration and shear tests. He also proposed the equation for pressure-sinkage relationship, given by



nc zkbk

zp )()( φ+= (11)

where p is pressure, is the width of a rectangular contact area, b z is sinkage, is the soil cohesive modulus, is the soil frictional modulus and

is the exponent of soil deformation. The value of , , and can be obtained from empirical test. From the experimental observations[13], the range between unloading and reloading can be approximated by a linear function in the pressure-sinkage relationship.

ck φk

n ck φk n

)()( zzkpzp uuu −−= (12)

where p and z are the pressure and sinkage, respectively, during unloading

or reloading; and are the pressure and sinkage, respectively, when unloading begins; and is the average slope of the unloading-reloading line. The slope of the unloading-reloading represents the degree of elastic rebound. If the slope is vertical, there is no elastic rebound. That means the terrain deformation is entirely plastic.

up uz

uk

The shear stress-shear displacement relationship proposed by Janosi and Hanamoto[13] is used for tangential shear forces, given by

)1)(tan(),( / Kjepczj −−+= φτ (13) where τ is the shear stress, p is the normal pressure, is the shear

displacement, and j

c φ are the cohesion and the angle of internal shearing resistance of the terrain, respectively, and K is the shear deformation modulus.

In summary, the proposed equations are applied for track system and soil interactions as;

Loading condition ( ) : pzz >

nc zkbkzp )()( φ+= (14)

)1)(tan(),( / Kjepczj −−+= φτ (15)

8-13

Unloading, Reloading condition ( ) : uz<z

ur zzzif <<

)()( zzkpzp uuu −−= (16)

)1)(tan(),( / Kjepczj −−+= φτ (17)

rzzif < 0)( =zp (18)

0),( =zjτ (19)

Loading condition after reloading ( ) uzz >

nc zkbkzp )()( φ+= (20)

)1)(tan(),( / Kjepczj −−+= φτ (21)

where is sinkage at the previous time step and is sinkage when the plastic effect of terrain is started during unloading. Figure 8 shows the simulation response to normal load of a track link on dry sand terrain when the vehicle is accelerated from the rest. The soil conditions for simulation are , , = ,

pz

95 kN

rz

1/.0 += nc mk 2/43.1528 += nmkNkφ c kPa04.1 φ = , and =1.1. The

pattern of result agrees to the experimental result shown in reference[13].

o28 n

uz

rz0.00 0.03 0.06 0.09 0.12 0.15 0.18

0.0

5.0x104

1.0x105

1.5x105

2.0x105

2.5x105

reloading

unloading

loading

Pres

sure

(N/m

2 )

sinkage (meter)

Figure 8. Simulation response to normal load of a dry sand terrain



Contact detect node

Contact segment area

Track link i

Figure 9. Mesh areas and detect nodes of a track link As shown in the Fig. 9, if the node of link i is in contact with triangular

patch ground, the contact force at the contact segment area can be computed using the equation as

j

)( segmentthjofareapF ijijp ×= (22)

)( segmentthjofareaF ijijs ×= τ (23)

where and are the normal pressure and shear stress, respectively.

Using the expression for the contact force as defined by the preceding equation, the contact force vector can be defined as

ijp ijτ

kij

skijp

ij FF tnF ˆˆ += (24) where and t are a unit normal vector and tangential vector of ground

patch . The virtual work of the ground force at a track link, which has the number of rectangle surface, is given by

kn

n

k

k

∑∑==

==n

j

ijTijn

j

iji WW11

Frδδδ (25)

where is a th node position vector of link i defined by inertia reference frame.

ijr j

8-15

8.6. SUMMARY AND CONCLUSION

For the sake of efficient component development of tracked vehicle at early

design stage, it is clearly proved that the multibody dynamic simulation methods can be very useful tool. The presented three dimensional multibody tracked vehicle consists of the hull, sprockets, road arms, road wheels, support rollers, and sophisticated suspension systems of hydro-pneumatic and torsion bars. A compliant force model is used to connect the rigid body track links. The tracked vehicle model has 191 bodies, 38 pin joints and 304 compliant bushing elements and has 956 degrees of freedom. The suspension, contact and bushing characteristics are extracted by empirical measurements and implemented into the simulation model. The efficient kinematic contact search algorisms between track system and chassis components are suggested and implemented. Two methods are developed for the interactions between track shoe body and ground. When the distributed node points on shoe body surface detect contact condition, direct forces are calculated based on the contact deformation on node points, or pressure and shear forces on each segment areas of the contact surface are calculated based on pressure-sinkage relationship and shear stress-shear displacement relationship. In order to validate and construct the simulation database, positions, velocities, accelerations and forces of the tracked vehicle are measured empirically. The simulation results show very good agreements with experimental measurements. Therefore, the suggested methods by using the multibody dynamic technologies can be used efficiently for tracked vehicle developments.



REFERENCES

[1] M. K. McCullough, and E. J. Haug, 1986, ''Dynamics of High Mobility Tracked

Vehicles'', ASME, Journal of Mechanisms Transmissions, and Automation in Design,

Vol.108, pp. 189-196.

[2] F.B. Huck, ''A Case for Improved Soil Models in Tracked Machine Simulation'',

Caterpillar, Inc.

[3] Tajima, and T. Nakanishi “Technical discussions” Komatsu Ltd.

[4] Nakanishi, T., and Shabana, 1994 "Contact Forces in the Nonlinear Dynamic analysis

of Tracked Vehicles", International Journal for Numerical Methods in Engineering,

Vol.37, pp. 1251-1275.

[5] Choi, J. H., Lee., H. C., Shabana, A. A., Jan. 1998, ''Spatial Dynamics of Multibody

Tracked Vehicles: Spatial Equations of Motion'', International Journal of Vehicle

Mechanics and Mobility, Vehicle System Dynamics, Vol. 29, pp. 27-49.

[6] Lee, H. C., Choi., J. H., Shabana, A. A., Feb. 1998, ''Spatial Dynamics of Multibody

Tracked Vehicles: Contact Forces and Simulation Results'', International Journal of

Vehicle Mechanics and Mobility, Vehicle System Dynamics, Vol. 29, pp. 113-137.

[7] R. Wehage, F. Huck “Technical discussions” Caterpillar Inc.

[8] H. S. Ryu, D. S. Bae, J. H. Choi and A. Shabana, 2000 ''A Compliant Track Model For

High Speed, High Mobility Tracked Vehicle'', International Journal For Numerical

Methods in Engineering, Vol. 48, 1481-1502.

[9] G. R. Doyle and G. H. Workman, 1979, ''Prediction of Track Tension when Traversing

an stacle'', Society of Automotive Engineers, 790416.

[10] A.G. Galaitsis, 1984, "A Model for Predicting Dynamic Track Loads in Military

Vehicles", ASME, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol.

106/289.

[11] R.M. Trusty, M.D. Wilt, G.W. Carter, D.R. Lesuer, 1988, ''Field Measurement of

Tension in a T-142 Tank Track'', Experimental Techniques.

[12] J. Choi, D. Park, H. Ryu, D. Bae, K. Huh, 2001 “Dynamic Track Tension of High

Mobility Tracked Vehicles” Proceedings of DETC’01, ASME Third Symposium on

8-17

[13] J. Wong, 2001, “Theory of Ground Vehicles” 3rd Ed. John Wiley & Sons.

Multibody Dynamics and Vibration, Pittsburgh, PA, USA.

[14] Shabana A. 1996 “Theory of Vibration, An Introduction, 2nd Ed.” Springer: New York.

[15] Berg, M., 1998 ''A Non-Linear Rubber Spring Model for Rail Vehicle Dynamics

Analysis'', International Journal of Vehicle Mechanics and Mobility, Vehicle System

Dynamics, Vol. 30, pp. 197-212.

[16] ADAMS Reference Manual, Mechanical Dynamics, 2301 Commonwealth Blvd, Ann

Arbor, MI 48105.

[17] Choi, J. H., 1996 “Use of Recursive and Approximation Methods in The Dynamic

Analysis of Spatial Tracked Vehicle”, Ph. D. Thesis, The University of Illinois at

Chicago.



9

AN EFFICIENT CONTACT SEARCH ALGORITHM FOR GENERAL MULTIBODY SYSTEM DYNAMICS

9.1. INTRODUCTION

This paper presents a contact analysis algorithm employing the relative coordinate system for the multibody system dynamics. Multiple-contact higher pairs are widely used in mechanical systems such as walking machines, feeding systems, driving chains, and tracks of off-road vehicles. Common design problems due to the multiple contacts among bodies are undercutting, jamming, backlash, and body interference.

The configuration space representation of a higher pair was proposed by Lozano-Perez [1] for robot motion planning. Sacks extended the configuration space concept in [2] for efficient detection of contact pairs. The relative position and orientation of a pair were mapped into the configuration space. The degrees of freedom of a pair became the dimension of the configuration space, which is divided into free space and contact space in the preprocessing stage of a dynamic analysis and is tabulated into a database. Run time query is made to decide whether a pair is currently in contact or not. When a higher pair has many degrees of freedom, formation of the configuration space and processing effort for a run time query may become extensive.

Wang presented an interference analysis method in [3]. Relative coordinates were defined for a contact pair and a kinematic closed loop including the contact pair was formed. Constraint equations arising from closed loops are solved for the relative coordinates including the ones for the contact pair. The canonical Hamiltonian formulation is used to derive a minimal set of dynamic equations of motion.

Mirtich proposed a contact detection algorithm consisting of narrow and broad phases in [4]. Candidate features are selected in the broad phase and contact



inspection is carried out in the narrow phase only among the candidate features. Haug presented a formulation for domains of mobility that characterizes

kinematic boundaries of multiple contact pairs in [5]. A surface-surface contact joint was developed by Nelson in [6]. Piecewise dynamic analysis method for a contact problem was employed in [7, 8]. Dynamic analysis is halted when a contact pair is detected to be in contact and is resumed with new velocities that are calculated from the momentum balance equations. One of drawbacks of this method is that too frequent halting and resuming of the numerical integration may occur when a contact pair toggles between contact and not contact status.

Zhong summarized many contact search algorithms in the area of the finite element analysis in [9]. All geometric variables necessary to detect a contact were expressed in the absolute Cartesian coordinate system. The penalty and Lagrange multiplier methods were proposed. The compliant contact model that is based on the Herzian law was used in [10]. Since the contact force is large and varied significantly, the differential equations of motion for this method are generally stiff.

A recursive formulation using the relative coordinates was proposed by Bae in Ref. [11]. The equations of motion were derived in a compact matrix form by using the velocity transformation method. The actual computation was carried out by using the recursive formulas developed for each joints. Realtime simulation of a vehicle system is carried out by the recursive method in Ref. [12]. The Jacobian matrix was updated once in while during time marching of the numerical integration. The recursive method was extended to the flexible body dynamics of constrained mechanical systems in Ref. [13]. A virtual body concept was employed to relieve the implementation burden of the flexible body dynamics coding. A compliant track link model was developed for tracked vehicles in Ref. [14]. A minimum set of the equations of motion was obtained by the recursive method. Concept of the configuration design variable with the recursive formulation was introduced in Ref. [15]. The recursive method is applied to efficiently detect a contact in this research.

This paper presents a hybrid contact detection algorithm of the configuration space method and bounding box method in conjunction with the compliant contact model. Two bodies of a contact pair are logically considered as a defense body on which the contact reference frame is defined and as a hitting body that

9-3

moves relative to the defense body, respectively. Contour of the defense body is approximated by many triangular patches which are projected on axes of the contact reference frame. Bounding box inside which contains base surface is divided into several blocks each of which is indexed on axis of the contact reference frame. Contact inspection for a contact pair is processed in the sequence of broad and narrow phases. Relative position vector of the hitting body to the defense body is projected on the axes of the contact reference frame and select candidate features that may come in contact shortly in the broad inspection phase, which greatly reduces the searching effort. It is not needed any database to be built prior to an analysis. Since the searching algorithm is coupled with stepping algorithm of the numerical integration, a strategy for deciding an integration stepsize is proposed. A numerical example is presented to demonstrate the validity of the proposed method.

9.2. KINEMATIC NOTATIONS OF A CONTACT PAIR

Consider a contact pair shown in Fig. 1. Two bodies of the contact pair will be referred as a hitting body and a defense body for convenience in the following discussions, respectively. The contours of the hitting and defense bodies will be referred as the hitting and target boundaries, respectively.

Figure 1 Kinematic notations of a contact pair



Figure 2 Contact reference frame and generalized coordinate

The coordinate system is the inertial reference frame and the primed coordinate systems are the body reference frames. The

orientation and position of the body reference frame is denoted by and , respectively.

ZYX −−

z′yx −′−′

A r

Double primed coordinate systems are the node reference frame of the hitting body and the surface and contact reference frames of the defense body, respectively. All geometric variables of the defense body are measured on the surface reference frame. The contact reference frame for the contact pair is defined on the left corner of the bounding box of the defense body, as shown in Fig. 2. The relative position and orientation of the hitting body to the defense body are defined as the generalized coordinates, which are denoted by and

as shown in Fig. 2. Therefore the generalized coordinates are directly used to detect a contact for the pair.

chd ′′

chA

9.3. DIVISION OF THE CONTACT DOMAIN

A surface-to-surface contact problem can be replaced by multiple sphere-to-surface contact problems. Therefore, the sphere-to-surface contact problem will be discussed in this research.

9-5

Contour of a smoothly shaped body has been represented by the 3D

NURBS(Non-Uniform Rational B-Spline)[16] in many commercial CAD programs. Since it is computationally extensive to find intersection lines or points between two surfaces, the defense surface is approximated by triangular patches and the boundary of the hitting body is represented by a set of spheres, as shown in Fig. 3. The numbers of patches and spheres must be decided by the degree of accuracy required.

Figure 3 Approximated defense and hitting surfaces

The bounding box of the defense surface in space can be divided into many

blocks each of which has a list of patches lying inside or on the block to efficiently process a contact detection, as shown in Fig. 4. Since the block locations are tabulated with respect to the contact reference frame attached to the defense body, they are constant. As a result, the locations do not needed to be calculated at every time steps, which significantly reduces computation time associated with the contact search.



Figure 4 Relationship patch and block: The patch, p belongs in the block, b

9.4. PRE-SEARCH

Every pairs of the boundary nodes of the hitting body and the patches on the defense body must be examined to detect a contact between two bodies, which is computationally extensive. In order to save the extensive computation, each node of the hitting body searches to find blocks of the contact domain to which it belongs in the pre-search stage, as shown in Fig. 5.

The relative position and orientation of the hitting body reference frame with respect to the contact reference frame shown in Fig. 2 can be directly available from the generalized coordinate d and . Therefore, the relative nodal position of the hitting body with respect to the contact reference frame is obtained as

ch′′ chA

nchchcn sAdd ′+′′=′′ (1)

where is the nodal position with respect to the hitting body reference frame. Direct comparison of the

ns′

cnd ′′ with this of the block locations of the defense body yields the state of a contact.

9-7

Figure 5 Node and blocks in pre-search stage

If a pair of a node and a block is in contact, post-search step will be proceeded.

The bounding box of the defense body is divided into many blocks. Each block has a list of patches lying within or on the block boundary. Therefore, the post-search step will be carried out only for the patches belonging to the blocks that have found to be in contact in pre-search step, as shown in Fig. 5.

9.5. POST-SEARCH AND COMPLIANCE CONTACT FORCE

The candidate patches on the defense surface have been selected for the post search step in the pre-search step. For the candidate patches, it is necessary to compute the amount of penetration to generate the contact forces, as shown in Fig. 6.

Figure 6 Node and patch in post-search stage



The relative position of a node with respect to the patch reference frame is obtained as follows.

pnd ′′

1pcnpn sdd ′′−′′=′′ (2)

where The vector pnd ′′′ is projected into the patch reference frame as

pnT

ppn dCd ′′=′′′ (3)

where is the orientation matrix of the patch reference frame with respect

to the contact reference frame. pC

The first step in the post search is to check whether the node is in contact with the patch or not by inspecting pnd ′′′ . In case of non-contact, the rest of procedures must be skipped. Otherwise, the penetration of the node into the patch is calculated by

pnT

p-rδ dn ′′′′′′= (4)

where is always positive. The δ pn ′′′ is a normal vector of a patch and a constant vector with respect to the patch reference frame.

Thus, the contact normal force is obtained by

32

1 mmmn δδ

δδckδf &&

&+= (5)

where k and are the spring and damping coefficients which are determined

by an experimental method, respectively and the is time differentiation of . The exponents m and generates a non-linear contact force and the exponent yields an indentation damping effect. When the penetration is very small, the contact force may be negative due to a negative damping force, which is not realistic. This situation can be overcome by using the indentation damping exponent greater than one.

c

δ& δ

1 2m

3m

9-9

The friction force is obtained by

nf fµf = (6)

where is the friction coefficient and its sign and magnitude can be determined from the relative velocity of the pair on contact position.

µ

9.6. KINEMATICS AND EQUATION OF MOTION FOR THE RECURSIVE

FORMULAS A contact search algorithm is proposed in the previous sections. The proposed

method makes use of the relative position and orientation matrix for a contact pair. This section presents the relative coordinate kinematics for a contact pair as well as for joints connecting two bodies.

Translational and angular velocities of the zyx ′−′−′ frame in the frame are respectively defined as

ZYX −−

wr&

(7)

Their corresponding quantities in the zyx ′−′−′ frame are defined as

≡

′′

=wArA

wr

Y T

T && (8)

where is the combined velocity of the translation and rotation. The

recursive velocity and virtual relationship for a pair of contiguous bodies are obtained in [17] as

Y

1)i(i1)i2(i1)(i1)i1(ii −−−− += qBYBY & (9) where denotes the relative coordinate vector. It is important to note that

matrices and are only functions of the . Similarly, the 1)i(i−q

1)i1(i−B 1)i2(i−B 1)i(i−q



recursive virtual displacement relationship is obtained as follows

1)i(i1)i2(i1)(i1)i1(iiδ −−−− += δqBδZBZ (10) If the recursive formula in Eq. (9) is respectively applied to all joints, the

following relationship between the Cartesian and relative generalized velocities can be obtained:

qBY &= (11) where is the collection of coefficients of the and B 1)i(i−q&

[ ]T 1ncTT

2T

1T0 ×= nY,,Y,Y,YY K (12)

[ ]T1nr

T)1(

T12

T01

T0 ×−= nnq,,q,q,Yq &K&&& (13)

where nc and nr denote the number of the Cartesian and relative coordinates,

respectively. Since in Eq. (11) is an arbitrary vector in q& nrR , Eqs. (9) and (11),

which are computationally equivalent, are actually valid for any vector such that

nrRx ∈&

xBX &= (14) and

1)i-(i1)i2-(i1)-(i1)i1-(ii xBXBX += (15)

where is the resulting vector of multiplication of and . As a

result, transformation of into is actually calculated by recursively applying Eq. (15) to achieve computational efficiency in this research.

ncRX∈ B xnrRx ∈ ncRBx∈

Inversely, it is often necessary to transform a vector in G ncR into a new

vector in GBg T= nrR . Such a transformation can be found in the generalized force computation in the joint space with a known force in the Cartesian space. The virtual work done by a Cartesian force is obtained as follows. ncRQ ∈

9-11

QΤZW δδ = (16)

where must be kinematically admissible for all joints in a system. Substitution of

Zδ

qBZ δδ = into Eq. (16) yields

*TTT δδδ QqQBqW == (17)

where . QBQ T* ≡

The equations of motion for constrained systems have been obtained as follows.

0)QλΦYMBF ΤΖ

T =−+= &( (18)

where the λ is the Lagrange multiplier vector for cut joints [18] in mR and represents the position level constraint vector in Φ mR . The and Q are the mass matrix and force vector in the Cartesian space including the contact forces, respectively.

M

The equations of motion and the position level constraint can be implicitly rewritten by introducing vq=& as

0λa,vqF =),,( (19)

0qΦ =)( (20) Successive differentiations of the position level constraint yield

(21) 0υvΦvqΦ q =−=),(&

0γvΦvvqΦ q =−= &&&& ),,( (22)


Equation (19) and all levels of constraints comprise the over determined differential algebraic system (ODAS). An algorithm for the backward differentiation formula (BDF) to solve the ODAS is given in [19] as follows.


0

βvvU

βvqU

vvqΦ

vqΦ

qΦ)λv,vq(F

xH =

++

++

)(

)(

)(

=

)β(

)β(

,,

,

,,

)(

20T0

10T0

&

&&&

&

&

(23)

where [ ]TTTTT λ,v,v,qx &= , , and are determined by the

coefficients of the implicit integrators and is an nr matrix such

that the augmented square matrix is nonsingular.

0β

qΦUT

0

1β 2β

0U m)(nr −×

The number of equations and the number of unknowns in Eq. (23) are the same, and so Eq. (23) can be solved for . Newton Raphson method can be applied to obtain the solution .

nx

nx

H∆xHx −= (24)

1,2,3,...i,i1i =+=+ ∆xxx (25)

0

0UU000UU0ΦΦΦ00ΦΦ000Φ

FFFF

H

T0

avq

vq

q

qqqq

x =

=

T00

T00

T0

ββ

&&&&&&

&&

(26)

Recursive formulas for and in Eq. (24) are derived to evaluate them

efficiently. xH H

9.7. NUMERICAL INTEGRATION STRATEGY

The sufficient condition for a successful numerical integration step is to satisfy both accuracy and stability of the state variables for a system without

9-13

contact. Satisfaction of the accuracy and stability is not sufficient for a system with a contact. Suppose a bullet collides with an object. If the object is thin, the bullet passes through the object without noticing it. If the object is thick and a moderately large step size satisfies both the accuracy and stability, the bullet penetrates too deep

at the first step of a contact. Large and sudden contact force due to the large penetration generally introduces a large numerical error in the state variables. The large numerical error often causes the integration step to fail. Therefore, the contact condition must be considered in deciding an integration step. In order to make a system transition from a non-contact status to a contact status smooth as much as possible, time of contact must be predicted accurately. However, the computationally extensive search algorithm must be triggered to predict the exact time of a contact even though two bodies of a contact pair are located in a distance. Easy and practical solution to this problem is to use the method of backtracking.

Figure 7 Buffer radius of a node

This paper adopted the concept of buffer radius shown in Fig. 7. In post-search stage, if no nodes with radius in the hitting body is contacted with the candidate lines in the defense body and some nodes with buffer radius are contacted, the integrating step will be decreased.

9.8. NUMERICAL EXAMPLE

The proposed algorithm is implemented in the commercial program RecurDyn.



A paper-feeding problem of a copying machine is solved to demonstrate the efficiency and validity of the proposed method.

Figure 8 Copying machine

The system has 255 degree of freedom and consists of five roller pairs and one

paper shown in Fig. 8. Each roller pair is modeled by using two driving rollers, two idlers, two driving bars, two idler bars, six joints and one nip spring. The paper is modeled by using 40-segmented bodies and 28 plate force elements. The segmented paper bodies and the roller pairs are contacted and it is modeled by using 160 sphere to surface contacts.

The paper goes through a path while contacting the roller pairs. The angular velocity of each driving roller reaches 10 rad/sec during one second. The tangential velocities of a driving roller and a leading segment body of the paper are shown in Fig. 9. The static and dynamic friction coefficient is 0.5 and 0.3, respectively.

The analysis was performed on an IBM compatible computer (PIII-933Mhz) and took about 260 sec. per 1 sec. for simulation. A copying machine is solved to demonstrate the effectiveness of the proposed algorithm

9-15

Figure 9 Tangential velocities of a driving roller and a leading segment body of paper

on a contact point

9.9. CONCLUSIONS This research proposes an efficient implementation algorithm for contact

mechanisms. The contact domain is divided into many blocks each of which contains the list of patches inside it. The search process consists of pre-search and post search steps. In the pre-search step, the bounding box technique is employed to find approximate contact state. Once the contact is detected in the pre-search step, the detailed contact condition is further examined in the post-search step. The compliance contact model is used to generate the contact force which is applied to the hitting and defense bodies. The relative coordinate formulation is used to generate the equations of motion. The local parameterization method is used to solve the differential algebraic equations. The integration stepsize is automatically reduced when a contact is expected soon. The proposed algorithm is implemented in the commercial program RecurDyn and a copying machine example is successfully solved.



REFERENCES

1. Lozano-Perez, T., "Spatial Planning: A Configuration Space Approach", IEEE

Transactions on Computers, Vol. C-32, IEEE Press, 1983.

2. Sacks, E. and Joskowicz, L., "Dynamical Simulation of Planar Systems with Changing

Contacts Using Configuration Spaces", "Journal of Mechanical Design", Vol. 120, pp.

181~187, 1998.

3. Wang, D., Conti, C. and Beale, D., "Interference Impact Analysis of Multibody Systems",

"Journal of Mechanical Design", Vol. 121, pp. 121-135, 1999.

4. Mirtich, B. V., "Impulse-based Dynamic Simulation of Rigid Body Systems", Ph. D thesis,

University of California, Berkeley, 1996.

5. Haug, E. J., Wu, S. C. and Yang, S. M., "Dynamic mechanical systems with Coulomb

friction, stiction, impact and constraint addition-deletion, I: Theory", "Mech. Mach.

Theory", Vol. 21(5), pp. 407-416, 1986.

6. Nelson, D. D. and Cohen, E., "User Interaction with CAD Models with Nonholonomic

Parametric Surface Constraints", Proceedings of the ASME Dynamic Systems and

Control Division, DSC-Vol. 64, pp. 235-242, 1998.

7. Wang, D., Conti, C., Dehombreux, P. and Verlinden, O., "A Computer-aided

Simulation Approach for Mechanisms with Time-Varying Topology", "Computers and

Structures", Vol. 64, pp. 519-530, 1997.

8. Wang, D., "A Computer-aided Kinematics and Dynamics of Multibody Systems with

Contact Joints", Ph. D Thesis, Mons Polytechnic University Belgium, 1996.

9. Zhong, Z. Z., "Finite Element Procedures for Contact-Impact Problems", Oxford

University Press, 1993

10. Lankarani, H. M., "Canonical Impulse-Momentum Equations for Impact Analysis of

Multibody System", ASME, "Journal of Mechanical Design", Vol. 180, pp. 180-186,

1992

11. Bae, D. S., Han, J. M., and Yoo., H. H., “A Generalized Recursive Formulation for

Constrained Mechanical System Dynamics”, “Mech. Struct. & Mach.”, Vol. 27(3), pp.

9-17

12. Bae, D. S., Lee, J. K., Cho, H. J., and Yae, H., “An Explicit Integration Method for

Realtime Simulation of Multibody Vehicle Models”, “Computer Methods in Applied

Mechanics and Engineering”, Vol. 187, pp. 337-350, 2000.

293-315, 1999.

13. Bae, D. S., Han, J. M., Choi, J. H., and Yang, S. M., “A Generalized Recursive

Formulation for Constrained Flexible Multibody Dynamics”, “International Journal for

Numerical Methods in Engineering”, Vol. 50, pp. 1841-1859, 2001.

14. Ryu, H. S., Bae, D. S., Choi, J. H., and Shabana, A. A., “A Compliant Track Link Model

for High-speed, High-mobility Tracked Vehicles”, “International Journal for Numerical

Methods in Engineering”, Vol. 48, pp. 1481-1502, 2000.

15. Kim, H. W., Bae, D. S., and Choi, K. K., “Configuration Design Sensitivity Analysis of

Dynamics for Constrained Mechanical Systems”, “Computer Methods in Applied

Mechanics and Engineering”, Vol. 190, pp. 5271-5282, 2001.

16. Farin, G., "Curves and Surfaces for Computer-aided Geometic Design", Academic

Press, 1997.

17. Angeles, J., "Fundamentals of Robotic Mechanical Systems", Springer, 1997.

18. Wittenburg, J., "Dynamics of Systems of Rigid Bodies", B. G. Teubner, Stuttgart, 1977.

19. Yen, J., Haug, E. J. and Potra, F. A., "Numerical Method for Constrained Equations

of Motion in Mechanical Systems Dynamics", Technical Report R-92, Center for

Simulation and Design Optimization, Department of Mechanical Engineering, and

Department of Mathematics, University of Iowa, Iowa City, Iowa, 1990.



10

LINEARIZED EQUATIONS OF MOTION FOR MULTIBODY SYSTEMS WITH CLOSED LOOPS

10.1. INTRODUCTION

Linearization is an important tool in understanding the system behavior of a nonlinear system at a certain state. As an example, the eigenvalues of the linearized equations of motion are very useful information in developing control logics. Linearization of an unconstrained system is relatively easier than that of the constrained systems due to the algebraic constraint equations and corresponding Lagrange multipliers. This research proposes a linearization method for the constrained mechanical systems and compares the results with those obtained from other methods.

Sohoni [1] presented an approach for automatically generating a linearized dynamical model, which is derived from the nonlinear equations of motion. The Lagrange multiplier term was kept constant in the linearized equations of motion. The velocity and acceleration level constraints have not been considered in the resulting linearized equations of motion. Neuman symbolically generated the dynamic robot model by Lagrange-Euler formulation and linearized the dynamic model about a nominal trajectory [2]. Balafoutis presented a computational method for recursive evaluation of linearized dynamic robot model about a nominal trajectory [3]. The formulation was applied to the robot systems, which are unconstrained systems. This formulation was generalized by Gontier [4] for general unconstrained mechanical systems. Similar formulations have been developed by the variational approach in Refs. [5,6]. A recursive formulation using the relative coordinates was proposed by Bae in Ref. [7]. The equations of motion were derived in a compact matrix form by using the velocity transformation method. The actual computation was carried out by using the recursive formulas developed for each joints. Realtime simulation of a vehicle

RecurDyn™ / Solver THEORETICAL MANUAL


system has been carried out by the recursive method in Ref. [8]. The Jacobian matrix was updated once in a while during time marching of the numerical integration. The recursive method was extended to the flexible body dynamics of constrained mechanical systems in Ref. [9]. A virtual body concept was employed to relieve the implementation burden of the flexible body dynamics coding. A compliant track link model was developed for tracked vehicles in Ref. [10]. A minimum set of the equations of motion was obtained by the recursive method. Concept of the configuration design variable with the recursive formulation was introduced in Ref. [11].

The equations of motion for multibody systems are highly nonlinear with respect to the relative positions, velocities, and accelerations. The equations of motion are perturbed to obtain the linearized equations of motion. Since the equations of motion are highly nonlinear, their perturbation involves with many arithmetic operations for a multibody system consisting of many bodies and joints. In case of open loop systems which do not have any constraints, the equations of motion result in the ordinary differential equations whose partial derivatives with respect to the relative coordinates, velocities, and accelerations has been obtained by several different methods in Refs. [2,3,4]. In case of closed loop systems which have constraints, these method cannot be used directly any more due to the constraints and corresponding Lagrange multipliers.

One of the intuitive methods to handle the constraints is to directly express the equations of motion only in terms of the independent relative positions, velocities, and accelerations. In order to achieve this goal, the relative coordinates must be divided into the independent and dependent coordinates and the dependent coordinates, velocities, and accelerations must be directly expressed in terms of independent ones. However, the independent and dependent coordinates, velocities, and accelerations are tightly and nonlinearly coupled by the position, velocity, and acceleration level constraints and the equations of motion are implicit function of the coordinates, velocities, and accelerations. As a result, it is very difficult to directly express the dependent coordinates, velocities, and accelerations in terms of independent ones and consequently to express the equations of motion only in terms of the independent coordinates, velocities, and accelerations.

The null space of the constraint Jacobian is first pre-multiplied to the

10-3

equations of motion to eliminate the Lagrange multiplier and the equations of motion are reduced down to a minimum set of ordinary differential equations. The resulting differential equations are still functions of all relative coordinates, velocities, and accelerations. Since the coordinates, velocities, and accelerations are tightly coupled by the position, velocity, and acceleration level constraints, direct substitution of the relationships among these variables yields very complicated equations to be implemented. As a consequence, the reduced equations of motion are perturbed with respect to the variations of all coordinates, velocities, and accelerations, which are coupled by the constraints. The position, velocity and acceleration level constraints are also perturbed to obtain the relationships between the variations of all relative coordinates, velocities, and accelerations and variations of the independent ones. The perturbed constraint equations are then simultaneously solved for variations of all coordinates, velocities, and accelerations only in terms of the variations of the independent coordinates, velocities, and accelerations. Finally, the relationships between the variations of all coordinates, velocities, accelerations and these of the independent ones are substituted into the variational equations of motion to obtain the linearized equations of motion only in terms of the independent coordinate, velocity, and acceleration variations.

The proposed method is implemented in the commercial program RecurDyn. Vibration analyses of a four bar mechanism and a vehicle system are carried out to demonstrate the validity of the proposed method.



10.2. RELATIVE COORDINATE KINEMATICS

X

Z

Y

ir

iy′

ix′

iz′iO

Oif

ig

ih

Figure 1 Coordinate systems and a rigid body

Figure 1 shows the coordinate system fixed on a body . In the figure, the

frame is the body reference frame and the i

iii zyx ′−′−′ ZYX −− frame is the inertial reference frame. Point O is the origin of ZYX −− , point is the origin of

iO

iii zyx ′−′−′ , and is the position vector of from O . The , , and are unit vectors along the

ir iO if ig

ih x′ , iy′ , and iz′ axes, respectively. Orientation matrix of the body is given as

[ iiii hgfA = ] (1)

Velocities and virtual displacements of point in the XiO ZY −− frame are defined as (see Refs. [4-5])

=

i

ii ω

rY

& (2)

=

i

i

δπδr

δZ (3)

Their corresponding quantities in the iii zyx ′−′−′ frame are defined as

10-5

(4)

iTi

iTi

ωArA &

≡

′′

=′i

ii ω

rY

&

(5)

≡

′′

=i

Ti

iTi

i

i

πArA

πδrδ

δZδδ

Figure 2 Kinematic relationships between two adjacent rigid bodies

X

Z

Y

iy′

ix′

iz′

)1( −iisii )1( −d

)1(y −′i

)1(x −′i

)1(z −′i

iO)1( −iO

ir)1( −ir

ii )1( −s

A pair of contiguous bodies is shown in Figure 2. Body 1i − is assumed to

be an inboard body of body and the position of point is i iO

1)i(i1)i(i1)i(i1)(ii −−−− −++= sdsrr (6)

By using Eq. (5), the angular virtual displacement of body in its local reference frame is

i

(7) 1)i(i1)i(iT

1)i(i1)-(iT

1)i(ii δδδ −−−− ′+′=′ qHAπAπ

where is determined by the axis of rotation and is defined as ii )1( −′H 1)i(i−A

iT

1)(i1)i(i AAA −− = (8)

Taking variation of Eq. (6) yields



1)i(i1)i(i1)i(iT

1)i(i1)i(i1)i(i1)i(iT

1)(i1)i(iT

1)-i(i1)i(i1)i(i1)i(i1)i(iT

1)(i1)i(iT

i

δ)~)((

δ)~~~(

δδ

1)i(i−−−−−−−

−−−−−−

−−

′′+′+

′′−′+′−

′=′

−qHAsAdA

πAsAdsA

rAr

q

(9)

where symbols with tildes denote skew symmetric matrices comprised of their

vector elements that implement the vector product operation and denotes the relative coordinate vector.

1)i(i−q

Combining Eqs. (7) and (9) yields the recursive virtual displacement equation for a pair of contiguous bodies

1)i(i1)i2(i1)(i1)i1(iiδ −−−− +′=′ δqBZδBZ (10)

where

′−′+′−

= −−−−−

−−

−

I0AsAdsI

A00A

B )~~~( 1)i(iT

1)i(i1)i(i1)i(i1)i(i1)i1(i

1)i(i

1)i(iT

T

(11)

′′′+′

=

−

−−−−−

−

−−

−

1)i(i

1)i(i1)i(iT

1)i(i1)i(i1)i(i

1)i(i

1)i(i1)i2(i

~)(1)i(i

HHAsAd

A00A

B qT

T

(12)

It is important to note that matrices and are functions of only

relative coordinates of the joint between bodies 1)i1(i−B 1)i2(i−B

(i 1)− and . As a consequence, further differentiation of the matrices and in Eqs.

(11) and (12) with respect to other than yields zero. The virtual displacement relationship between the absolute and relative coordinates for the whole system can be obtained by repetitive application of Eq. (10) as

i

(i−1)i1(i−B

1)i

1)i2B

(i−q

qBZ δδ =′ (13)

where is the velocity transformation matrix with relationship between

Cartesian and relative coordinates. The relationship between Cartesian velocity B

10-7

Y q&′ and relative velocity can be derived in the same manner.

+′

=

M

=

qBY &=′ (14)


The variational form of the Newton-Euler equations of motion for a constrained multibody system is

0QλΦYMZ Ζ =−′ ′ )(δ TT & (15)

where and Q are the mass matrix and general force vector in Cartesian

space, respectively. M

Z′δ must be kinematically admissible for all joints except cut joints [12]. In the equation, and λ , respectively, denote the constraint equations and the corresponding Lagrange multiplier in in which m is the number of the constraint equations. Substituting the virtual displacement relationship and acceleration relationships into Eq. (15) yields (see Ref. [5])

ΦmR

qBqB &&&& +=Y&

n*T* R∈=−+ F0QλΦqMF q&& (16)

where n is the number of generalized coordinates and the mass matrix and force vector are defined as

*M*Q

BMBT=* (17)

)(* qBMQBQ T &&− (18) A recursive method has been proposed to compute Eqs. (17) and (18) in Ref.

[7].



10.4. ELIMINATION OF LAGRANGE MULTIPLIERS AND LINEARIZATION

OF THE EQUATIONS OF MOTION

The relative coordinates can be partitioned into dependent coordinates and independent coordinates q such that the sub-Jacobian Φ is well-conditioned. Variational form of the cut constraint equations can be written as

q Dq

I Dq

0qΦqΦΦ qq =+= ID δδδ

ID (19)

The can be obtained from Eq. (19) as Dδq

I1

D δδID

qΦΦq qq−−= (20)

By using the relationship in Eq. (20), is represented as I

1D δδ

IDqΦΦq qq

−−=

Iδδ qNq = (21) where

−=

−

IΦΦ

N qq ID

1

(22)

Direct calculation of shows that is the null space of as TT

qΦN N qΦ

[ ] 0ΦΦ

IΦΦΦNq

qqqq =

−= T

TT1-TTT

I

DDI

)( (23)

As a result, pre-multiplication of Eq. (16) by gives TN

0QNqMNF =−= *T*T* && (24)

where Lagrange multiplier λ term was eliminated since is the null space of Φ . N q

10-9

However, the equations of motion are dependent on not only the dependent variables ,

and but also independent variables , and q& . Taking variation of Eq. (24) yields

*

Fqδ*

[ ]

&*

q

q

q

+

+

=

δ

δ

δ

qqq

&&

&

δδδ

F q

q& q&& q q& &

D

D D I

q+&δ

Φ

0

+

=

I000

I

Fq&& δ*

q&&δ

000

I

0

Tq&δqδ

qFqF q =+= &&&δ ** (25)

Equation (25) can be rewritten in a matrix form as

0qqq

FFF qqq =

&&

&&&

δδδ

** (26)

Variations of position, velocity and acceleration level constraints are

0qΦΦ

qΦΦ

0Φ

qqq

qq

q

=&&&&

&&

δ

δ (27)

Appending the trivial identity relationships for the variations of independent

coordinates, velocities and accelerations to Eq. (19) yields

=

I

I

I

qqq

000

IΦΦΦ0ΦΦ00Φ

qqq

qq

q

&&

&&&&

&

δδδ

(28)

Equation (28) is solved for the q&&δ and substituted into the

linearized equations of motion in Eq. (26) to yield the following linearized equations of motion only in terms of the variations of independent coordinates, velocities and accelerations:



(29) [ ] 0qqq

I000000000

IΦΦΦ0ΦΦ00Φ

FFFqqq

qq

q

qqq =

−

I

I

I

1

***

&&

&&&&

&&&&

δδδ

Direct comparison of Eq. (29) and the following linearized equations of

motion yields the M)

, C)

and K)

matrices:

0qKqCqMFq

=++= III* δˆδˆδˆδ * &&& (30)


10.5.1. FOURBAR MECHANISM WITH A SPRING Figure 3 shows a four bar mechanism with a spring. The system consists of

four revolute joints and one spring and their material properties are defined in Table 1. As a result, three generalized coordinates, , and are defined for the first three revolute joints and the remaining one revolute joint is defined as a cut joint. The constraint equations are introduced from the cut joint.

1θ 2θ 3θ

2θ

1θ 3θ

Link

Link

Link 500

400 Cut joint

Figure 3 A four-bar mechanism with a spring

10-11

Table 1 Material property of bodies and a spring

Mass (kg) Inertia Moment (kg*mm^2)

Link A 7.707 161760.83 Link B 3.946 53005.79 Body Link C 7.707 161760.83

Stiffness (N/mm) Damping (N*sec/mm) Spring 10.0 0.0

Dynamic analysis of the mechanism is performed to obtain the time domain

response. FFT of the time response is performed to extract dominant frequency domain response. Figures 4 and 5 show the time and frequency responses, respectively.

The proposed linearization method is applied for the system. The dominant frequency and corresponding mode shape are shown in Figure 6 and Table 2. The frequency obtained from the proposed method and that obtained from FFT analysis of the time domain responses are shown to be very close, which validates the proposed method.

Figure 4 Angle of link C in time domain



Figure 5 Response in frequency domain

Figure 6 Mode shape of fourbar mechanism

Table 2 Undamped natural frequency and mode shape from the proposed method

Undamped Natural Frequency (Hz) Mode

1θ 2θ 3θ 5.040164E+00

5.773503E-01 -5.773503E-01 5.773503E-01

10-13

10.5.2. CANTILEVER BEAM DRIVEN BY A MOTION The system characteristics of a rotating cantilever beam differ from those of

beam in a static state, because the stiffness of the beam is changed by a centrifugal force due to the rotational motion. (see Ref. [13]). A cantilever beam rotating with the angular velocity is shown in Figure 7. ω

Figure 7 A rotating cantilever beam

Length of the beam is 6.8 m, density of the material is 14705.88 kg/m3, Young's modulus of the material is 7.0×108 N/m2. Area of the cross section is 0.002 m2, the moment of inertia 4.0×10-7 m4. The beam is divided into 21 lumped mass and 20 beam elements. Figure 8 shows the lowest three natural frequencies of the rotating beam. As the angular speed increases, the bending natural frequencies are shown to be increased.

Figure 8 The relationship between angular velocity and natural frequencies



10.5.3 A SPRING SYSTEM WITH 2 D.O.F.

A spring model shown in Fig. 9 is a system with two D.O.F, and the system has two masses, joints and spring elements. Their material properties, spring and damping coefficients are shown in Table 3.

Figure 9 A spring model

Table 3 Material properties, spring and damping coefficients

Mass1 5 Kg Mass2 3 Kg

Length of m1 300 mm Spring coefficient (k1) 10 N/mm Spring coefficient (k2) 20 N/mm

If the rotational angle θ is small, θθ ≅sin and the equation of motion of this system can be derived as:

0=

+

+

ykkl

klklkl

ymI θθθ

22

22

2

12

2

2

240

0&&

&& (31)

From Table 3, Eq. (31) can be replaced as:

(32) 020000300030001350

30015.0

=

+

yyθθ

&&

&&

10-15

The characteristic equation of this spring system is derived from Eq (32).

03200003000

300015.01350=

−−

λλ

(33)

Also, the analytic natural frequencies can be computed as:

)Hz(76.17fsec)/rad(6.11112455

)Hz(019.9fsec)/rad(66.563211

22

11

=⇒==ω

=⇒==ω (34)

Finally, the eigenvalues of this spring system is validated shown in Table 4.

Table 4 Eigenvalues of spring model

Undamped Natural Frequency (Hz) Mode number RecurDyn/Eigenvalue Analytic solution

1 9.01862E+00 9.019 2 1.77621E+01 17.76

10.5.4 A CANTILEVER BEAM

Two cantilever beam models shown in Figs. 10 and 11 have a fixed-free end

condition and ten lumped masses. One is modeled by using ten beam force elements and the other is modeled by using one flexible body of RecurDyn. The flexible beam model is originally generated in ANSYS. The material properties and geometry conditions of the beam are shown in Table 4.

Figure 10 Beam model using RecurDyn/Beam element



Figure 11 Beam model using RecurDyn/Flexible body element

Table 5 The material properties and geometry conditions of beam Length 0.4 m Mass 3.9888 Kg

Young’s modulus

9101× N/m2

Inertia of area 810215.1 −× m4 Area 0.0018 m2

In Ref. [14], the analytic natural frequencies of these beams are computed as:

42

1 875.1ALEIρ

ω = , 42

2 694.4ALEIρ

ω = , 42

3 855.7ALEIρ

ω = (35)

By replacing Eq. (35) with Table 5, the natural frequencies can be computed as:

8601.32537.24875.1 42

1 =⇒== nfALEIρ

ω

1929.240085.152694.4 42

2 =⇒== nfALEIρ

ω

7477.676714.425855.7 42

3 =⇒== nfALEIρ

ω

10-17

Finally, the eigenvalues of this beam model is validated shown in Table 6.

Table 6 Eigenvalues of cantilever beam model Undamped Natural Frequency (Hz) Mode

number Beam element Flexible Body Analytic solution 1 3.84002E+00 3.84259E+00 3.8426 2 2.37455E+01 2.38154E+01 23.8154 3 6.55744E+01 6.60152E+01 66.0152 4 1.26483E+02 1.28016E+02 128.016 5 2.05481E+02 6 2.65264E+02

In addition, RecurDyn can show the mode shapes of the beam model through 3D animation, as shown in Figs. 12 and 13.

(a) 1st mode shape (b) 2nd mode shape

(c) 3rd mode shape

Figure 12 The mode shapes of model using RecurDyn /Beam element



(1) 1st mode shape (2) 2nd mode shape

(3) 3rd mode shape

Figure 13 The mode shapes of model using RecurDyn/Flexible body

10.6. CONCLUSIONS

In this paper, a linearization method for constrained multibody systems is proposed for the non-linear equations of motion employing the relative coordinates. Null space of the constraint Jacobian is pre-multiplied to the equations of motion to eliminate the Lagrange multipliers and to reduce the number of equations. The set of differential equations are perturbed in terms of all relative positions, velocities and accelerations. The position, velocity and acceleration level constraints are perturbed to express the variations of all relative positions, velocities and accelerations in terms of the variations of independent positions, velocities and accelerations, which are substituted into the perturbed equations of motion. The equations of motion perturbed with respect to the , and q& finally become the corresponding equations perturbed with respect to the , and . Eigenvalues and eigenvectors are then computed from the equations of motion perturbed with respect to the , and . The proposed method is implemented in a commercial program RecurDyn. Numerical results obtained from the proposed method are in good agreement with the results reported in the literature and obtained by other methods.

q q& &

q&Iq I Iq&&

Iq Iq& Iq&&

10-19

REFERENCES

1. Sohoni VN, Whitesel J. Automatic Linearization of Constrained Dynamical

Models. ASME. Journal of Mechanism, Transmission, and Automation in Design,

Vol. 108, pp 300-304, 1986.

2. Neuman CP, Murray JJ. Linearization and Sensitivity Functions of Dynamic

Robot Models. IEEE Transactions on Systems, Man, and Cybernetics, Vol. SMC-

14, No.6, pp.805-818, 1984.

3. Balafoutis CA, Misra P, Patel RV. ecursive Evaluation of Linearized Dynamic

Robot Models. IEEE Journal of Robotics and Automation, Vol.RA-2, No.3,

pp.146-155, 1986.

4. Gontier C, Li Y. Lagrangian Formulation and Linearization of Multibody System

Equations. Computers & Structures, Vol.57. No.2, pp.317~331, 1995.

5. Bae DS, Haug EJ. A Recursive Formulation for Constrained Mechanical System

Dynamics: Part I, Open Loop Systems. Mech. Struct. and Machines, Vol. 15, No.

3, pp.359-382, 1987.

6. Bae DS, Haug EJ. A Recursive Formulation for Constrained Mechanical System

Dynamics: Part II, Closed Loop Systems. Mech. Struct. and Machines, Vol. 15,

No. 4, pp. 481-506, 1987.

7. Bae DS, Han JM, Yoo HH. A Generalized Recursive Formulation for

Constrained Mechanical System Dynamics. Mech. Struct. & Mach., Vol. 27(3),

pp. 293-315, 1999.

8. Bae DS, Lee JK, Cho HJ, Yae H. An Explicit Integration Method for Realtime

Simulation of Multibody Vehicle Models. Computer Methods in Applied


9. Bae DS, Han JM, Choi JH, Yang SM. A Generalized Recursive Formulation for

Constrained Flexible Multibody Dynamics. International Journal for Numerical



Methods in Engineering, Vol. 50, pp. 1841-1859, 2001.

10. Ryu HS, Bae DS, Choi JH, Shabana AA. A Compliant Track Link Model for

High-speed, High-mobility Tracked Vehicles. International Journal for

Numerical Methods in Engineering, Vol. 48, pp. 1481-1502, 2000.

11. Kim HW, Bae DS, Choi KK. Configuration Design Sensitivity Analysis of

Dynamics for Constrained Mechanical Systems. Computer Methods in Applied


12. Wittenburg J. Dynamics of Systems of Rigid Bodies. B. G. Teubner Stuttgart,

1977.

13. Southwell R, Gough F. The Free Transverse Vibration of Airscrew Blades.

British A.R.C. Reports and Memoranda No. 766, 1921.

14. L. Meirovitch, “ Analytical Methods in Vibrations”, MACMILLAN, 1967.

10-21



11

NONLINEAR DYNAMIC MODELING OF SILENT CHAIN DRIVE

11.1. INTRODUCTION

Chain drives are widely used in the power transmission applications in the automotive field for a long time because they are capable of transmitting large power at high efficiency and low maintenance cost. However, the noise and vibrations created by chain drives have always been major problems, especially for higher speed, lighter weight, and higher quality. Noise and vibrations in chain systems are largely caused by chordal(polygonal) action and impacts between chain and sprocket. The links of the chain form a set of chords when wrapped around the circumference of the sprocket. As these links enter and leave the sprocket, they impart a jerky motion to the driven shaft by chordal action. The chordal action causes chain span longitudinal and transverse vibrations. Whereas, impact between sprocket and link excites high frequency vibration and is a major source of noise in chain drives at high speeds. In order to minimize such problems, silent chains are introduced in many camshaft drives of motorcycle/automobile engines and the primary drive between the engine and transmission, as well as in other high-speed applications. It is also used with the object of increasing chain life. However, in spite of the widespread use of silent chain drives, surprisingly little works have been published about their dynamic analysis. This may be due to three major difficulties; the first is the complexity of the contact algorithms among components, the second is small integration step size resulting from the impulsive contact forces and the use of stiff compliant elements to represent the joints between the chain links, and the third is the large number of the system equations of motion to solve.


EFFICIENT CONTACT AND NONLINEAR DYNAMIC MODELING OF AUTOMOTIVE SILENT CHAIN DRIVE

Camshaft Sprocket (Idler Sprocket)

Crankshaft Sprocket

Pivot Guide

Tensioner

Fixed Guide

Figure 1. Silent Chain Drive Model of Automotive Engine

Chen and Freudenstein [1] presented a kinematic analysis of chain drive mechanism with the aim of obtaining insight into the phenomena of chordal action, with the associated impact and chain motion fluctuation. Veikos and Freudenstein [1] developed a lumped mass dynamic model based on Lagrange’s equations of motion and showed chain drive dynamics and vibrations. Wang [3, 4] investigated the stability of a chain drive mechanism under periodic sprocket excitations and studied the effect of impact intensity in their axially moving roller chains. Kim and Johnson [5, 6] developed a detailed model of the roller-sprocket contact mechanics that allowed the first determination of actual pressure angles and a multi-body dynamic simulation. This investigation is based on Kane’s dynamic equations. Choi and Johnson [7, 8] investigated the effects of impact, polygonal action, and chain tensioners into the axially moving chain system and showed the transverse vibration of chain spans. Quite recently Ryu et

11-3

al [9] developed very detailed chain models including contact forces for links, sprockets and idlers with special application to large-scaled civilian and military tracked vehicles. There has been some design analysis in the view of dynamic behaviors of silent chain in powertrain industry and commercial software [10]. However they showed some primitive dynamic analysis and design of silent chain system because it has high frequency contact forces, speedy revolution and large number of bodies.

The purpose of this work is to investigate and suggest the dynamic modeling and analysis of silent chain drive mechanism with high speed revolution using multibody dynamic techniques. In this investigation, numerical skills of multibody chain dynamic analysis are employed and showed very good agreement of physical phenomenon of silent chain system. Dynamic tension, impact forces, and vibration of chain links are explored for the sake of understanding dynamic behaviors of the chain system.



11.2. MULTIBODY MODELING OF SILENT CHAIN DRIVE

As shown in Fig. 1, in general a chain drive mechanism has four main

components, which are sprockets, chain links, guides and tensioner element. The sprockets can be recognized as drive sprockets and idler sprockets. The chain link can includes link plates, guide plates, and pins. The tensioner element maintains stable tension during operation by adjusting pressure force to the chain link system. While roller chain mechanism has engagements between pins and sprocket, since silent chain mechanism engages between chain link teeth and sprocket teeth, there is much less chodal vibrations and can transmit the power more quietly. In this investigation the dynamic analysis and numerical modeling techniques are presented by using multibody methods.

11.2.1 SPROCKET

The sprockets of the chain system are interacted by the introduced contact and friction forces acting on between the chain and the sprocket teeth. The crank sprocket of the system is driven by motion constraint. This motion constraint can be constant or time dependent. In this investigation the sprocket is modeled as a rigid body and attached on ground by revolute joint. The geometry of sprocket teeth profiles consists of a series lines and arcs with different length and radii as shown in Figure 2. The sprocket of silent chain is shaped more like a gear than one of roller chain.

Figure 2. Geometry of Silent Chain Sprocket

11-5

11.2.2 SILENT CHAIN LINK

Roller chains, although having excellent wear and strength capability, are inherently noisy and oscillatory. As a result, inverted-tooth chain mechanisms were developed in order to reduce the forcing function of the noise-producing mechanism. The difference in noise performance between silent and roller chains can be attributed to the manner in which they engage and disengage the sprocket teeth. After the sprocket tooth initially contacts the chain link, and as the engagement proceeds, a combination of rolling and sliding motion occurs between the tooth and link contacting surfaces. Such an engagement mechanism effectively spreads the engagement time over a significant interval, thereby minimizing tooth/link impact and its inherent noise generation.

A silent chain consists of several layers of links connected with pins. Since there is no advantage for the modeling of pins and multi layer links as separate components, in this investigation these multi layer links are treated as a rigid body with mass and inertia property which takes into account the effects of the pins. An individual silent chain link looks much different comparing to a roller chain link. The geometry of link profile, which resembles a tooth, consists of several lines and arcs in a complex arrangement as shown in Fig. 3. As used in the roller chain from previous work, the connections between links are modeled with bushings to account for the flexibility in this investigation. Though the sprockets of the silent chain serve in the same function of the rolling chain system, however, they are designed to engage specifically with the links of the silent chain with different tooth contour as illustrated.


Figure 3. Components of Silent Chain Link System


11.2.3 TENSIONER AND CHAIN GUIDE

In a chain drive system, the chain guide ensure that the chain remains on the path, while tensioner try to keep constant tension of chain system. Usually the chain guide directs the tight chain portion which runs from the driven sprocket to the driving sprocket. Conversely, the chain arm directs the slack portion of the chain which runs opposite (from the driving to the driven). The pivot guide also serves to distribute the force on the chain from the hydraulic tensioner to maintain certain level of chain tension. In this investigation hydraulic tensioning force model is used which is offered from hydraulic tensioner manufacturer.

The chain guide and the chain arm are both modeled as separate rigid body parts. The geometric profiles of the guides consist of a series arcs with different radii. If desired, the chain guides can be modified so that they are constructed as flexible bodies for the calculation of vibrations, stresses and bending moments, etc.

11.2.4 EQUATIONS OF MOTION AND INTEGRATION

Since the chain system interacts with the frame component through the contact forces and adjacent chain links are connected by compliant force elements, each chain link in the chain system has six degrees of freedom which are represented by three translational coordinates and three Euler angles. The equations of motion of the frame structure such as sprockets that employs the velocity transformation defined by Choi [9] are given as follows :

)( ri

ri qBMQBqMBB TT &&&& −= (1)

where and B are relative independent coordinates and velocity transformation matrix of the engine chassis subsystem, and M is the mass matrix, and Q is the generalized external and internal force vector of the frame structure subsystem, respectively. Since there is no kinematic coupling between the frame structure subsystem and chain subsystem, the equations of motion of the chain subsystem can be written simply as

riq

11-7

ttt QqM =&& (2)

where , and Q denote the mass matrix, the generalized coordinate and force vectors for the chain subsystem, respectively. Consequently, the accelerations of the frame structure components and the chain links can be obtained by solving Eqs. (1) and (2).

t qM t t

Many different types of integration methods can be employed for solving the equations of motion for mechanical systems. Explicit methods have small stability region and are often suitable for smooth systems whose magnitude of eigenvalues is relatively small. Contrast to the explicit methods, implicit methods have large stability region and are suitable for stiff systems whose magnitude of eigenvalues is large. In the model used in this investigation, a contact between two bodies is modeled by compliance elements. Lumped characteristics of the spring and damper must represent elastic and plastic deformations, and hysterisis of a material. Such characteristics may include artificial high frequencies which are not concern of a design engineer. Unless such artificial high frequency is filtered, an integration stepsize must be reduced so small that integration can’t be completed in a practical design cycle of a mechanical system. To achieve this goal, the implicit generalized-alpha method [9, 11] has been employed to filter frequencies beyond a certain level and to dissipate an undesirable excitation of a response in this investigation. One of the nice advantages of the generalized-alpha method is that the filtering frequency and dissipation amount can be freely controlled by varying a parameter in the integration formula. As a result, the generalized-alpha method is the most suitable integration method for integrating the equations of motion for stiff mechanical systems.



11.3. CONTACT FORCE ANALYSIS

The contact collision algorithms for a silent chain drive used in this investigation are composed of three main routines such as search routines for, sprocket teeth and chain link contact, chain guide and chain link contact, and side guide of chain link and sprocket contact. The contact positions and penetration values are defined from the kinematics of components in searching routines. Thereafter a concentrated contact force is used at the contacted position of the contact surface of the bodies. A detailed discussion on the formulation of the contact collision is represented in this section, respectively. Efficient search algorithms should be considered seriously because there are large number of chain link bodies and sprocket which take long time to search all the bodies whether they are in contact or not.

11.3.1 STRATEGE OF CONTACT SEARCH

For the efficient search of the sprocket-chain link contact kinematics, the contact search algorithm is divided by pre-search and post-search. In the pre-search, bounding circle relative to sprocket center is defined. All of chain links are employed to detect a starting link and ending link which has a possibility of sprocket contact. Then, chain links from starting link are investigated the engagement with sprocket valley. Post-search means a detailed contact inspection for chain links in a bounding circle. Once a starting and ending link is found at one time through pre-search prior to analysis, only detailed search is carried out by using the information of starting link and ending link from the next time step. There are four contact possibilities such as, arc-line, arc-point, arc-arc and line-point contact for interaction between the sprocket teeth and chain link.

11-9

11.3.2 LINE-ARC CONTACT

X

Y

Z

iX

iY

iZ

itX

itY

itZ

iR

itu

jpX jY

jZ

jR

jpu

p

t

ijku

aθ

jpY

jX

Link arc

Tooth line

Chain link coordinate system

Sprocket coordinate system Global coordinate system

Tooth coordinate system

Figure 4. Line-Arc Contact Kinematics

The contact conditions between the sprocket teeth line segment and the chain link arc segment can be determined. A coordinate system is attached to each of the sprocket surfaces shown in Fig. 4. The surfaces of the tooth line are approximated by plane surfaces and the axis of each surface coordinate system is assumed to be parallel to the tooth surface. The surfaces of the chain link arc segment are approximated by plane surfaces and the axis of each arc

origin coordinate system is assumed to be directed to the starting arc point from arc origin. The orientation of the tooth surface coordinate system with respect to the global system is defined by

it

it

it ZYX

jpX

itX

k

ik

iit AAA = (3)


where is the transformation matrix that defines the orientation of the coordinate system of the sprocket and is the transformation matrix that

defines the orientation of the tooth line surface coordinate system with respect to the sprocket coordinate system. The orientation of the link arc coordinate system with respect to the global system is defined by

iAi i

kA

k it

it

it ZYX


jl

jja AAA = (4)

where is the transformation matrix that defines the orientation of the chain arc surface coordinate system with respect to the chain link coordinate system.

jlA

l jp

jp

jp ZYX

The global position vector of the coordinate system of the tooth surface is defined as

k

it

iiit uARr += (5)

where iR is the global position vector of the coordinate system of the sprocket and i i

tu is the position vector of point with respect to the origin

of the sprocket coordinate system .

tiii ZYX

The global position vector of the center of the chain link arc segment, denoted as point p , can be defined as

jp

jjjp uARr += (6)

where jR is the global position vector of the origin of chain link j , is

the transformation matrix of chain link

jA

j and jpu is the position vector of

point p defined in the chain link coordinate system . jjj ZYX

The position vector of the center of the arc of chain link j with respect to the origin of the tooth line surface coordinate system can be defined in the global coordinate system as

it

jp

ijk rru −= (7)

The components of the vector along the axes of the tooth line surface coordinate system are determined as

ijku

[ ] ijk

Tit

Tijz

ijy

ijx

ij uuu uAu == (8)

Necessary but not sufficient conditions for the contact to occur between the chain link arc and the sprocket tooth line surface are k

kijx lu ≤≤0 (9)

11-11

pt

ijzpt wwuww +≤≤−− (10)

rijy ≤u (11)

where is the length of the tooth line surface , is half width of the

tooth and is half width of the chain link outer plate and kl

w

k tw

p r is the radius of

the chain link arc. If the above conditions are satisfied, it has to be checked if contact point is existed in the arc range for the next step.

gd −=jik , where (12) ][ hgfA =i

k

[ ] jik

Tja

Tjiz

jiy

jix

jik ddd dAd == (13)

),(atan2 jix

jiyk dd=θ (14)

ak θθ ≤≤0 (15)

where is the opposite signed normal vector of the tooth line surface ,

is the angle of with respect to the link arc segment coordinate system

and is the angle of arc segment.

jikd k

kθjikd

aθ

If the above conditions are satisfied, the penetration is evaluated as ijδijy

ij ur −=δ (16)



11.3.3. ARC-POINT CONTACT

X

Y

Z

iX

iY

iZ

itXi

tY

itZ

iR

itu

jY

jZ

jR

jpu

pt

ijku

aθjX

Link point

Tooth arc

Chain link coordinate system

Global coordinate system

Tooth coordinate system

Sprocket coordinate system

Figure 5. Arc-Point Contact Kinematics

There are two arc-point contact possibilities such as convex arc vs. point and concave arc vs. point contact for arc-point interaction between the sprocket teeth and chain link. Figure 5 shows a convex arc-point contact kinematics. The arc-point contact conditions between the sprocket teeth and the chain link can be determined. A coordinate system is located at the center point of the sprocket arc surfaces.

it

it

it ZYX

The position vector of the point p of chain link j with respect to the center point of the tooth surface coordinate system can be defined in the global coordinate system such as in Eqs. (7) and (8)

Necessary but not sufficient conditions for the contact to occur between the chain link point and the sprocket tooth surface are k

ruu ijy

ijx ≤+ 22 )()( (17)

ptijzpt wwuww +≤≤−− (18)

where r is the radius of the sprocket arc segment, is half width of the tw

11-13

tooth and is half width of the chain link outer plate. pw

If the above conditions are satisfied, it has to be checked if contact point is existed in the arc range for the next step.

),(atan2 ijx

ijyk uu=θ (19)

ak θθ ≤≤0 (20)

where is the angle of with respect to the sprocket arc segment

coordinate system and is the angle of arc segment. kθ

ijku

aθ

If the above conditions are satisfied, the penetration is evaluated as ijδ

22 )()( ijy

ijx

ij uur +−=δ (21)

11.3.4. ARC-ARC CONTACT

There are four arc-arc contact possibilities such as convex vs. convex, convex vs. concave, concave vs. convex, concave vs. concave arc contact for arc-arc interaction between the sprocket teeth and chain link. Since the radius and angle of each arc are given at geometry, the contact kinematics between arcs can be calculated by expanding arc-point contact logic. At the center of the arc a marker is attached and X axis is fixed to the starting point of arc. The monitoring vector between arc centers can be easily detected whether they are in contact boundary or not using the arc angles with respect to the X axis of the marker. If the vector is in contact boundary and the length between the centers of arcs is less than the sum of the radii of arcs, they are considered in contact situation.

11.3.5 LINE-POINT CONTACT

The search kinematics of line-point contact is one of the most simple search algorithms in contact analysis. An axis of marker can be attached on the line and the vertical vector from the point to line can be evaluated whether the point is in contact with line, respectively.



11.3.6 CONTACT FORCE MODEL

In the field of multi-body dynamics, one of the most popular approximation of the dynamic behavior of a contact pair has been that one body penetrates into the other body with a velocity on a contact point, thereafter the compliant normal and friction forces are generated between a contact pair. In this compliant contact force model, a contact normal force can be defined as an equation of the penetration, which yields

(22) δδδ &nmn ckf −−=

where δ and are an amount of penetration and its velocity, respectively. The spring and damping coefficients of and c can be determined from analytical and experimental methods. The order of the indentation can compensate the spring force of restitution for non-linear characteristics, and the order n can prevent a damping force from being excessively generated when the relative indentation is very small. As it happens, the contact force may be negative due to a large negative damping force, which is not realistic. This unnatural situation can be resolved by using the indentation exponent greater than one. A friction force can be determined as follows.

δ&

km

nf fvf )(µ= (23)

11.4. NUMERICAL STUDY OF AN AUTOMOTIVE SILENT CHAIN SYSTEM

Four cylinder DOHC (double overhead cam) engine valve drive mechanism is employed for the sake of numerical verification of proposed methods as shown in Figure 1. A silent chain drive system has 1 crankshaft sprocket, 2 camshaft sprockets, 1 fixed guide, 1 pivot guide, tensioner element, and 135 chain links. The crank sprocket of the system is rotated by motion constraint. Resistance torque is applied at each camshaft sprockets. Hydraulic tensioning force model is used which is offered from manufacturer.

Figure 6 shows the computer simulation model of automotive silent chain

11-15

system in computer graphic environment. The system consists of 143 rigid bodies, 270 bushing force elements to connect chain link bodies, 4 revolute joints, 2 resistance torque and a hydraulic force element of tensioner. It has 815 degrees of freedom.

Figure 7 and 8 demonstrate the trajectory and velocity of the chain link during the cycle around the system when the engine runs 4000 rpm, respectively. The X-Y trajectory of the links agrees the defined path of the chain motion and the magnitude of link velocity with respect to system inertia reference frame reflect the linear velocity of 4000rpm as clearly shown in Fig. 8. Figure 9 shows the contact force between a chain link and the sprockets or the chain guides and figure 10 shows the dynamic chain tension measured between chain links during simulation. Since the hydraulic auto tensioner is attached on guide arm, the dynamic tension of the chain is controlled not to have excessive or be loosened. Dynamic analysis of the silent chain system is performed for 200 milli-sec. It is found that the CPU simulation times is 4039 sec on a Pentium 1.8 GHz platform personal computer. Note that since the numerical results from the proposed methods are almost showing the real physical behaviors and dynamic characteristics of the chain mechanism, the proposed methods using multibody dynamic techniques can be valid and suitable for the design of the silent chain system, accordingly.

Figure 6. Simulation Model of Automotive Silent Chain System



Figure 7. Trajectory of the Chain Link

Figure 8. Velocity of the Chain Link

11-17

Cam Sp. Cam Sp. Crank Sp.Cam Sp.

Pivot Guide

Pivot Guide

Fixed Guide

Figure 9. Contact Forces of the Chain Link at 4000 rpm

Figure 10. Dynamic Tension of the Chain Link at 4000 rpm



11.5. FUTURE WORK AND CONCLUSIONS

It is clearly proved in this investigation that using the multibody dynamic simulation methods the dynamic analysis of silent chain mechanisms can be achieved clearly. While previous works showed rough estimations of the silent chain system, the proposed methods in this paper show the possibility of the replacement of real prototype at early design stage. The presented three dimensional silent chain consists of the driving sprocket, idle sprockets, pivot guide, fixed guide, tensioner, and chain links. Pre and post contact search algorithms are employed in order to increase the simulation speed significantly. For the sprocket teeth and link teeth, guide and link contacts, line-arc, arc-point, arc-arc, and line-point kinematic interactions are presented in this investigation. A compliant force model is used to connect the rigid body chain links. The silent chain model has 143 bodies, 4 pin joints, tensioner element and 270 compliant bushing elements and has 815 degrees of freedom. The numerical study of automotive silent chain system shows that the tendency of the chain motion and tensions are close as real system and it shows the characteristics of silent chain comparing to roller chain with less oscillation.

11-19

REFERENCES

1. C. K. Chen and F. Freudenstein, ''Towards a More Exact Kinematics of Roller

Chain Drives”, ASME Journal of Mechanisms, Transmission, and Automation in

Design, Vol.110, No.3, 123-130 (1988)

2. N. M. Veikos and F., Freudenstein, "On the Dynamic Analysis of Roller Chain

Drives: Part1 and 2", Mechanism Design and Synthesis, DE-vol 46, ASME, NY,

431-450 (1992)

3. K. W. Wang, ''On the Stability of Chain Drive Systems Under Periodic Sprocket

Oscillations'', ASME Journal of Vibration and Acoustics, Vol. 114, 119-126

(1992)

4. K. W. Wang, et al, ''On the Impact Intensity of Vibrating Axially Moving Roller

Chains'', ASME Journal of Vibration and Acoustics, Vol. 114, 397-403 (1992)

5. M. S. Kim and G. E. Johnson, Advancing Power Transmission into the 21st

Centrury, DE-vol. 43-2, ASME, NY, 689-696 (1992)

6. M. S. Kim and G. E. Johnson, Advances in Design Automation, DE-vol. 65-1

(B. J. Gilmore et al., eds), ASME, NY, 257-268 (1993)

7. W. Choi and G. E. Johnson, Vibration of Mechanical Systems and the History of

Mechanical Design, DE-vol. 63 (R. Echenpodi et al., eds), ASME, NY, 29-40

(1993)

8. W. Choi and G. E. Johnson, Vibration of Mechanical Systems and the History of

Mechanical Design, DE-vol. 63 (R. Echenpodi et al., eds), ASME, NY, 19-28

(1993)

9. H. S. Ryu, D. S. Bae, J. H. Choi and A. Shabana, ''A Compliant Track Model For

High Speed, High Mobility Tracked Vehicle'', International Journal For

Numerical Methods in Engineering, Vol. 48, 1481-1502 (2000)

10. “Phased Chain System Quietly Transmits Power”, Automotive Engineering,



Dec. (1995)

11. J. Chung, J. M. Lee, ''A New Family of Explicit Time Integration Methods for

Linear and Non-linear Structural Dynamics'', International Journal for

Numerical Methods in Engineering, Vol.37, 3961-3976 (1994)

12

DYNAMIC ANALYSIS AND CONTACT MODELING FOR TWO DIMENSIONAL MEDIA

TRANSPORT SYSTEM

12.1. INTRODUCTION

Recently the media transport systems, such as printers, copiers, fax, ATMs, cameras, film develop machines, etc., have been widely used and being developed rapidly. Especially, in the development of those systems, the media feeding mechanism for paper, film, money, cloth etc., is an important key technology for the design and development of the media transport systems. Tedious and iterative experimental trial and errors methods have been essential way to determine kinematic mechanisms of parts dimensions, and materials, etc for the media machine developers. Since the iterative trial & error methods are truly inefficient, in order to shorten the time, reduce the cost, and improve the machine performance, it has been absolutely required to develop the computer simulation tool, which analyses the paper feeding and separation process.

Cho and Choi [1] developed a computational modeling techniques for two dimensional film feeding mechanisms. The flexible film is divided by several thin rigid bodies which are connected by revolute joints and rotational spring dampers. The primitive computer implementation methods for contact search algorithms are presented. Diehl [2, 5] presented the local static mechanics of electrometric nip system for media transport system. The nonlinear finite element method and experimental measurement techniques are used to investigate the large deformable rollers. Several unique phenomena, such as skewing sheet, etc., of nip feeding system are well described in this research. Ashida [3] suggested the computer modeling techniques for the design and analysis of film feeding mechanisms. The primitive dynamic analysis of two dimensional film feeding models are presented by using commercial computer


DYNAMIC ANALYSIS AND CONTACT MODELING FOR TWO DIMENSIONAL MEDIA TRANSPORT SYSTEM

program. The paper feed mechanism with friction pad system is investigated by Yanabe [4] by using commercial nonlinear FEA program. It show the local separation phenomenon between papers and roller, and proved very good agreement with experimental measurements. Shin [6, 7] developed web simulation and design tools using roll tensions. They show that the control of tensions of each segment is the key design factors for web system.

In this investigation, a numerical modeling method and dynamic analysis of the two dimensional flexible sheet for thin flexible media materials such as paper, film, etc., and their roller and guide contacts are suggested by using multibody dynamic techniques. Since the flexible sheet undergoes large deformation with assumed linear material properties, the flexible sheet has been modeled as a series of thin rigid bars connected by revolute joints with rotational spring dampers force elements. It shows good visual appearance of the sheet under severe bending conditions. An efficient contact search and force analysis between sheet and rollers, and guides are developed and implemented numerically. The sheet is fed by contact and friction forces when it contacts with rotating rollers or guides. In order to detect a contact phase efficiently, the bounding box method is used in this contact search algorithm. The method has an advantage that the number of contact search can be smaller than conventional methods for a system in which the position of rollers and guides are fixed on a point of a base body. The proposed numerical models for media transport systems will make it possible to confirm the potential problems of jamming by given different sheet size, weight, stiffness, temperature, humidity extremes, sheet velocity due to misalignment of drive-driven roller sets, and roller velocities due to gap, wear or etc.

12-3

12.2. TWO DIMENSIONAL FLEXIBLE MULIBODY SHEET

In general, there are two methods to build a thin 2-D flexible sheet for

dynamic analysis. One is to employ beam element at discretized sheet body, and the other is small rigid bar interconnected by revolute joint with rotational spring-damper forces. In this investigation, the second method is used and proposed the modeling techniques.

Figure 1 Modeling definition of a two dimensional flexible sheet

Several research works show that the most efficient way to model two-dimensional approximation of the proper behavior of a sheet can be a series of rigid bars connected by revolute joints and rotational spring-dampers as shown in Figure 1 [1, 3]. The sheet is divided into a number of rigid bars with mass. The mass and inertia moment of each rigid bar can be defined as follows

ss tLm ρ= (1)

12)( 22

sszz

LtmI += (2)

where, ρ is a sheet density per unit depth, is thickness, and is length

of each rigid bar. The leading body is connected to a ground by a planar joint to guarantee an in-plane motion. The planar joint has one rotational and two translational degrees of freedom. The body is connected to the body by a revolute joint and rotational spring damper. The revolute joint has one rotational degree of freedom between two rigid bars. The relative angle of

st sL

(ii )1+

)1( +iiθ



is directly integrated. The torque of the rotational spring-damper is computed as following

)1()1( ++ −−= iiii ck θθτ & (3)

s

s

LtEk

12

3

= (4)

kc ξ= (5)

where, )1( +iiθ and are relative angles and angular velocities of the

revolute joints, and )1( +iiθ&

E and ξ are the young’ s modulus and the structural damping ratio of a sheet.

Figure 2 Contact geometry of two-dimensional sheet

The contact geometry of a sheet is described as a box and two circles as shown Figure 2. The x-axis of the body reference frame of each rigid bar is defined along longitudinal length direction and the y-axis is defined by right hand rule. The mass center of each rigid bar is located at the center point of box. In order to generate a continuous contact force, two circles are located on both sides of the box. Even thought the proposed assumed method for flexible sheet has an excellent visual appearance of the sheet under severe bending conditions, this approach shows the lack of continuity between rigid bodies, which can cause noise problems when the sheet is contact with rollers. It has also rigid leading and trailing effect of the sheet. Problems can be overcome with introducing a circular edge at leading and trailing points of each rigid bar.

There can be another approach to assume flexible sheet in dynamic analysis,

12-5

which employs a series of beam forces, and for the contact definitions, a rigid bar can be attached simply. One of the advantages of this approach is a natural definition of the flexible properties using the beam elements. However this approach can cause problems with the contact definitions since it has possible gaps and the lack of continuity between rigid contact bodies. The contact forces on the edges of the rigid bodies are amplified as torques applied where the rigid body is connected to the junction of two beams, and the rigid leading and trailing edges of the sheet cause unnatural behaviors.

12.3. CONTACT FORCE ANALYSIS

In the field of multi-body dynamics, one of the most popular approximation of the dynamic behavior of a contact pair has been that one body penetrates into the other body with a velocity on a contact point, thereafter the compliant normal and friction forces are generated between a contact pair. Figure 3 shows the schematic diagram of contact force analysis used in this investigation.

Figure 3 Contact forces between a contact pair

In this compliant contact force model, a contact normal force can be defined as an equation of the penetration [9], which yields



(6) δδδ &nmn ckf −−=

where δ and are an amount of penetration and its velocity, respectively.

The spring and damping coefficients of k and can be determined from analytical and experimental methods. The order of the indentation can compensate the spring force of restitution for non-linear characteristics, and the order can prevent a damping force from being excessively generated when the relative indentation is very small. As it happens, the contact force may be negative due to a large negative damping force, which is not realistic. This unnatural situation can be resolved by using the indentation exponent greater than one. The phenomenon is very important for the case of sheet contact interaction since it is very thin and light. A friction force can be determined as follows.

δ&

cm

n

nf fvf )(µ= (7)

where, and nf )(vµ are a contact normal force and a friction coefficient, respectively.

12.3.1. KINEMATICS NOTATIONS

The coordinate system is the inertial reference frame and the single primed coordinate systems are the body reference frames, and the double primed coordinate system is the contact reference frame in order to define contact conditions as shown in Figure 4. The orientation and position of the body reference frame are denoted by and , respectively.

YX −

A r

12-7

Figure 4 Kinematic notations of a contact pair

12.3.2. SHEET AND ROLLER INTERACTIONS

In this investigation, two kinds of rollers are defined for the system. One is a fixed roller with one rotational degree of freedom. The fixed roller is linked to the ground with a revolute joint. The other is a movable roller, which has two degrees of freedom for a translational and a rotational motion. The movable roller is linked to rotational axis retainer (RAR) with a revolute joint and the retainer is linked to the ground with a translational joint. The contact geometry of rollers is described as a circle as shown in Figure 5

Figure 5 Definition of rollers



Two different interactions between roller and sheet are introduced in this

investigation. Since the proposed flexible sheet is constructed by linear part and circular part, these are interactions between linear part and rollers, and circular part and rollers, as clearly illustrated in Figure 6

Figure 6 Sheet and roller interaction

In the case of linear part contact with rollers, the contacted penetration is

determined as follows:

rR−= ysr,dδ , (8) )r(rAd srT

ssr −=′

where, is the orientation matrix of a rigid bar, and is the radius of a

contacted roller, respectively. The location of contact between rigid bar and roller can be defined as follows:

sA rR

′′

=′

02/)( sys tsign sr,

xsr,

c dd

s , (9)

and )ds(AAs srscs

Trrc ′−′=′ (10)

where, and t are the orientation matrix of a roller and the thickness of

the sheet. The relative velocity at the contact point can be determined as rA s

12-9

( )

( )scrsrcrr

rsrcrr

swArswA

sArsAr

′′−−′′+

′−−′+

s

dt~~ &r

Tn

Tn

ru

u

=

=

d

&

&δ (11)

cTn du &= (12)

and tangential relative velocity is

c

T du &ttv = (13)

where, and rw′ sw′ are the angular velocities of a roller and a rigid bar with

respect to each body reference frame, and u and are the normal and

tangent vectors of relative position between rigid bar and roller, respectively. n tu

12.3.3. ROLLERS INTERACTIONS

A circle to circle contact is used to describe the interactions between circular rollers. In this circle to circle contact, the positive normal direction is same in the direction of the relative position vector between two roller center points. The tangent direction vector is determined by the right hand rule. The relative velocity and the contact forces at the contact point can be computed similarly as the sheet and roller interactions.

12.3.4. SEEET AND GUIDE INTERACTIONS

Guide has three types. Commonly used sheet guides for media transport machines can be divided into three different types, which are an arc guide with radius and angle, a linear guide with two points, and a circle guide similar to a roller. In order to avoid the complex contact detect algorithms. It is assumed that the arc and line guide are interacted with the circular part of rigid bars of the sheet. However, in the case of circle guide, both linear and circular parts of the



sheet are interacted with.

Figure 7 Sheet and arc guide interactions

As shown in Figure 7, the relative displacement between a circular edge of

rigid bar and arc guide can be determined as

gggsssgs sArsArd ′−−′+= (14)

where, and are the center position and the orientation of the guide,

and the vectors of gr gA

gs′ and ss′ are positions of the arc reference frame and the

circular edge center position with respect to each body reference frame, respectively. If the vector is projected into the arc reference frame, the

resultant vector can be represented as follows gsd

gsgs dCAd gg

T)(=′′ (15)

where, is the orientation matrix of the arc reference frame. The relative

angle between x-axis of the arc reference frame and the resultant vector of Eq. 15 is within an arc angle, which can be written as

gC

gθ≤′′

′′′′≤ − )(cos0 1

gs

gT

gs

dfd (16)

12-11

where, is the arc angle and is a constant unit vector of . If

the condition of Eq. 16 is satisfied, the penetration between circular part of sheet and arc can be defined as follows

gf ′ 1′ [ ]T00gθ

gsgs Rt −+′′= dδ (17)

where, is a radius of the arc guide. The contact positions can be computed

as follows. gR

ngg R us c ′′−=′′ (18)

cggT

c

cc

sCAAs

dss

sg

ss

gsgsg

′′=′

′′−′′=′′ (19)

where, is the normal direction vector and determined nu ′′

gs

gs

dd

un ′′

′′=′′ (20)

The tangent direction vector is determined by the right hand rule, and the

relative velocity at the contact point is defined as follows.

)(~)(~

)()((

gcggggssss

gcgggsssdtd

sCswArsswAr

sCsArssArd

gsc

gscc

′′+′′−−′+′′+=

′′+′−−′+′+=

&&

& (21)

where, and gw ′ sw′ is the angular velocities of guide and a bar with respect

to each body reference frame, respectively. The contact forces at the contact point can be computed similarly as described in the sheet and roller interactions.



Figure 8. Sheet and line guide interactions

The sheet and line guide interactions are clearly illustrated in Figure 8. If the component of the vector d defined in the double primed line guide

reference frame is the range of guide length, simple circle and line contact algorithm is used in this investigation. After definitions of penetration and its derivative, the contact force is created to restitute each body as similar as previous interactions between sheet and guides.

x gs

12. 4. EQUATIONS OF MOTION

Figure 9 Kinematic relationships between rigid bars of the sheet

12-13

Since the multibody sheet system interacts with the roller and guide

components through the contact forces and adjacent rigid bars are connected by revolute joint and rotational spring damper forces as shown in Figure 9, each sub-rigid bar in the sheet system has one degree of freedom which is represented by one rotational coordinates and the leading body has three free coordinates. The equations of motion of the sheet system that employs the velocity transformation defined by Bae [8] are given as follows:

)( r

ii qBMQBqMBB TT &&&& −= (22)

where , and are relative independent coordinates, velocity transformation matrix, and Cartesian velocities of the media feeding system, and

is the mass matrix, and is the generalized external and internal force vector of the media feeding system, respectively. The velocity transformation matrix of the sheet is more explicitly as

riq B q&

M Q

B

=

−−−− 1)n2(n2321)n1(n1221)n1(n0121)n1(n

232122231012121231

122012121

012

BBBBBBB

0BBBBBB00BBB000B

B

LLL

MMMMM

L

L

L

where the recursive velocity and virtual relationship for a pair of rigid bars are

obtained [8] as

1)i(i1)i2(i1)(i1)i1(ii −−−− += qBYBY & (23)

and denotes the relative coordinate vector. It is important to note that

matrices and are only functions of the . 1)i(i−q

(iB 1)i1− 1)i2(i−B 1)i(i−q



12. 5. NUMERICAL RESULTS

The proposed algorithm is implemented and a film-feeding problem is solved to demonstrate the efficiency and validity of the proposed method.

Figure 10 Film feeding machine

The system has 29 degrees of freedom and consists of four fixed rollers and three movable rollers, five line guides, one arc and circle guide and one sheet of film shown in Figure 10. The sheet is modeled by using 20 rigid bars. The density and Young’ s modulus of sheet are 2.2e-6( kg ) and 2250( ), respectively. And the thickness and length of sheet are 0.5( mm) and 200( ), respectively.

3/ mm 2/ mmN

mm

Figure 11 Slip between rollers and sheet

12-15

The film goes through a path while contacting the roller pairs. The

circumferential speed of each driving roller is 300( ). The slip velocities between driving rollers and the sheet are shown in Figure 11. The path of first, second and third segment bodies of the thin film are plotted as shown in Figure 12. The x and y axes of the plot are displacements measured in the directions of x and y axes in the global reference frame, respectively.

sec/mm

Figure 12 Path of segmented bodies of film

The analysis was performed on an IBM compatible computer (PIII-933Mhz)

and took about 60 sec. per 1 sec. for simulation.



12. 6. CONCLUSIONS

The dynamics and modeling techniques of two-dimensional media transport system is investigated in this paper. The flexible sheet is divided by finite number of rigid bars. Linear motions are constrained in order to allow rotations between the rigid bars of the sheet. Rotational spring damper force is applied for the reflection flexible stiffness of the sheet. From previous empirical measurements in manufacturing process effective stiffness and damping coefficients are substituted in this investigation. Compliant contact force model is used for the interactions between sheet rollers, and guides. Kinematics notations of the contact search algorithms for the media transport system are clearly represented. A simple film feeding example is represented in this investigation and manufacture [3] confirms that simulation results have very good agreement with experimental measurements. The media transport system manufactures have rely on trial error techniques for the design of their core mechanisms, however the proposed method by employing multibody dynamics in this paper can reduce many difficulties at the early design stage.

12-17

REFERENCES

1. H. J. Cho, and J. H. Choi, 2001, “2DMTT development specification” Technical

report, FunctionBay Inc.

2. Ted Diehl, 1995, “Two dimensional and three dimensional analysis of nonlinear

nip mechanics with hyper elastic material formulation” Ph. D. Thesis,

University of Rochester, Rochester, New York

3. Tsuyoshi Ashida, 2000, “The meeting material of The Japan Society for

Precision Engineering” Japan

4. http://www.yanabelab.nagaokaut.ac.jp

5. http://www.me.psu.edu/research/bension.html

6. http://www.engext.okstate.edu/info/WWW-WHRC.htm

7. Shin, K. H., 1991, “Distributed Control of Tension in Multi-Span Web Transport

Systems “, Ph. D. Thesis Oklahoma State Univ.

8. D. S. Bae, J. M. Han, and H. H. Yoo, 1999, “A Generalized Recursive

Formulation for Constrained Mechanical System Dynamics”, Mech. Struct. And

Machines, Vol. 27, No 3, pp 293-315

9. Lankarani H. M. and Nikravesh P. E., 1994, “Continuous Contact Force Models

for Impact Analysis in Multibody Systems”, Journal of Nonlinear Dynamics,

Kluwer Academic Publishers, Vol. 5, pp 193-207


http://www.me.psu.edu/research/bension.html

http://www.engext.okstate.edu/info/WWW-WHRC.htm


13

HYDRAULIC AUTO TENSIONER (HAT) FOR BELT DRIVE SYSTEM

13.1. INTRODUCTION

The hydraulic auto tensioner is a device that automatically adjusts the tension for engine belt drive system. By reducing the noise due to play that occurs if the tension on the belt drive system is insufficient and by holding the tension constant, an auto tensioner extends the product life of the belt drive system and is an indispensable part for improving engine reliability [1]. It is important to analyze and to predict the dynamic behavior and the characteristics of the hydraulic auto tensioner for design of the system. At this, numerical simulation models can provide significant advantages in early design stage referred in [2] and [3]. A simple simulation technique of HAT is applied for the initial design of belts and chains using commercial multibody software [7]. Figure 1 shows the hydraulic auto tensioner system. The plunger is connected to the belt drive system. The spring force and the hydraulic force of the pressure chamber create the damping force and are balanced with the load that is from belt drive system. The check ball has the function of the check valve for control the oil flow through orifice between the plunger and the cylinder.



Plunger

Cylinder

Check Ball

Pressure Chamber

Plunger

Spring

Check Ball

Spring

Figure 1. Hydraulic auto tensioner system

Figure 2 shows the schematic diagram of the operating principle of HAT. As

the tension of the belt drive system decreases, and the pressure of chamber decreases, the check ball moves down and the check valve opens. Afterward due to the plunger moves up by the spring force and the plunger pushes the belt, and then the tension of belt system is increased. As the tension of the belt drive system increases, the plunger moves down by the load and the plunger pushes the pressure chamber, and it leads that the pressure of chamber increases, finally the check ball moves up and the check valve closes. As a result, the oil flows through the leakdown and the plunger moves down slowly.

Tension Decreasing Tension Increasing

Oil

Flow

Figure 2. Operating principle

13-3

Since the tension of the belt drive system is oscillated over 200~300Hz, the hydraulic auto tensioner must be a reciprocating hydraulic device that can respond to frequencies up to 300Hz.

The multibody simulation model of the hydraulic auto tensioner is presented in the following sections. The differential equations are used to describe the function and damping characteristics of the hydraulic auto tensioner, and the circle to curve contact model is used for the movement of the check ball. In this investigation, the developed HAT model is tested numerically for multibody belt drive system.

13.2. MULTIBODY SIMULATION MODEL

The hydraulic auto tensioner consists of cylinder, plunger and check ball. The spring force and the damping force of the plunger relative to the cylinder balance these bodies. The spring force is built up by the spring preload and the spring rate multiplied by the spring stiffness. The damping force is a friction force and a hydraulic force that is proportional to the relative velocity of plunger and cylinder [3].

The schematic diagram of analysis model is shown in Figure 3. When the plunger is loaded from belt drive system, the spring force and the hydraulic force react against the motion of the plunger. The hydraulic force from the check ball is ignored in this investigation since it is relatively small amount. The motion of the plunger is assumed to have the parallel direction to the motion of the cylinder. The check ball has the spring force and the hydraulic force from the plunger. The motion of the check ball is also assumed to have the parallel direction to the motion of the plunger. The check ball is contacted between plunger and retainer.



Tra

nsla

tion Jo

int

Sprin

g F

orc

e

Hydra

ulic

Forc

e

Tra

nsla

tion Jo

int

Check

ball

Sprin

g F

orc

e

Hydra

ulic

Forc

e

Plunger

Conta

ct

Cylinder

Belt Drive System

Joint

Figure 3. Schematic diagram of analysis model

11.3. THE EQUATION OF MOTION

When the external load is forced to the plunger, the equation of the plunger motion is following [3].

loadriccp

cpphydraulicppp

FfxxxxKFxm

−−−

−−=

)sgn()(.

&&

&& (1)

where, , , and are the displacement of the plunger, and its first and px px& px&&

13-5

second time derivatives, and , , , and are the mass of plunger, the displacement of the cylinder, its first time derivative, and the stiffness coefficient

of the plunger spring, and , , and are the hydraulic force, the friction force and the load form belt drive system, respectively.

p cx

hydraulic

B

K

F

−

m Kc

ric

B −

x&

f

B

hydraulic

x

p

loadF

BO

B

F

x

−

pF .

BB x −−=&&

)

(

(. &

oil&

cx& )p −p x⋅ &( V+ &S=

In the case of check ball, since its motion is forced by the hydraulic force, the spring force, and contact force, the equation of motion of the check ball can be written as [3]

contactp

cB

Fx

xm

+

)&η (2)

13.4. HYDRAULIC FORCES

The hydraulic forces that interact with the check ball and the plunger are obtained from the pressure of the pressure chamber. The pressure is caused by the volume variation of the pressure chamber and the oil flow rate. The volume variation of the pressure chamber can be described by relative velocity between plunger and cylinder. The rate change of the chamber volume is given by the following equation [5].

pBairchamber QQVV ++=& (3)

where is the compressed volume rate of pure oil and is the compressed volume rate of air component in the oil. Q

oilV& airV&

p is leak oil flow rate out of the high compression chamber at high pressure phase, QB is the oil flow rate through check valve, and Sp is the effective area of hydraulic force, respectively.



13.4.1. OIL FLOW RATES THROUGH THE CHECK VALVE

Accordingly to the check ball moves between the plunger and the retainer, the check valve opens or closes and the oil flows. When the check valve opens, the oil flow through the check valve is shown in Figure 4.

Po

Pi

d

r

α

QB

Figure 4. Oil flow rate through check valve

As the resistance of the oil through the check valve depends on the orifice area, in this investigation, the dynamic resistance is considered for the turbulent flow of the oil flow through the check valve [4], which yields,

ioiodB PPgPPACQ −⋅−⋅⋅=γ2)sgn(

(4)

where, is discharge coefficient of check valve, A is the orifice area, g is

gravity acceleration, and γ is weight density of oil, respectively. QdC

B represents the oil stream flowing rate through the opened check valve into or out of the pressure chamber. The area of orifice is obtained such as

ααπ cossin2 drA = (5)

13-7

13.4.2. OIL FLOW RATES THROUGH THE LEAK

BETWEEN PLUNGER AND CYLINDER

As the pressure of chamber is different comparing to the air pressure, the oil flow through the gap between plunger and cylinder is shown in Figure 5. The oil flowing between the plunger and the cylinder is laminar flow. The oil speed is faster than the plunger speed. As shown in Figure 5, variation of the oil speed is fully depended on the pressure difference between the pressure chamber and reserver, and it is not affected by the plunger speed. The oil flow rate between the gap of plunger and cylinder, Qp , can be written as [4]

)(12

2 3

ioP

p PPlhrQ −=

µπ

(6)

where µ is the coefficient of viscosity of oil.

Po

Pi Qp

l

rp h

Figure 5. Oil flow rate through leak As shown in the Figure 4 and 5, we can consider about the relationship

between the plunger speed and the flow rate. It is assumed that inflow does not induce any outflow from the pressure chamber by considering compressibility,



and expansion and compression processes are isentropic. It is also assumed that there is no cavitation caused by negative pressure. The volume of air in chamber is obtained as.

0.

1

airi

oair VP

PV κ

= (7)

where κ is the ratio of specific heat and Vair.0 is the initial air volume. Since air can be compressed, the volume rate is achieved by using the following equation, such as

ii

airair P

PVV &&κ−

= (8)

, and the volume of oil in chamber can be approximated by

pcpoil SxxV ⋅−≈ )( (9) In the case of oil, since it can also be compressed with high pressure, the

volume rate of oil is written as.

ioil

oil PKVV && −

= (10)

where K is the bulk modulus.

The equations (4), (6), (8) and (10) are substituted into the equation (3), and

the differential equation for the pressure of the camber can be obtained, accordingly;

13-9

−−

−−+−+−

=

KV

PV

PPgPPACPPlhrxxS

oil

i

air

ioioioP

CPP

i

κ

γµπ

α2)sgn()(

122)(

3

&&

p&

)

(11) The hydraulic force to the plunger and to the check ball yield as

)(. iophydraulicp PPSF −⋅= (12)

)(. ioBhydraulicB PPSF −⋅= (13)

where SB can be obtained from Figure 4 as following.

( 2cosαπ ⋅= rSB (14)



13.5. CONTACK OF THE CHECK BALL The contact analysis of the check ball employs the circle to curve contact

method [6] in this investigation. This method is very efficient algorithm in contact detection and force generation of the check ball contact.

Figure 6. Concept of circle to curve contack The candidate lines on the plunger body have been selected for the contact of

the check ball. For the candidate lines, it is necessary to compute the amount of penetration to generate the contact forces, as shown in Figure 6.

The relative position of a check ball with respect to the contact reference

frame is obtained as follows. pnd ′′

1pcnpn sdd ′′−′′=′′ (15)

where the vector pnd ′′′ is projected into the contact reference frame as

13-11

pnd ′′T

ppn Cd =′′′ (16)

where Cp is the orientation matrix of the contact reference frame. The penetration of the node into the patch is calculated by

pnT

p- dn ′′′′′′= rδ (17)

where δ is always positive. The pn ′′′ is a normal vector of a line and a constant

vector with respect to the contact reference frame. Thus, the contact normal force is obtained by

321 mmmcontact ckF δδ

δδδ &&

&+=

(18)

where k and c are the spring and damping coefficients which are determined by assumed numerical experiences, or experimental methods, respectively and the

is time differentiation of . The exponents and generates a non-linear contact force and the exponent yields an indentation damping effect. When the penetration is very small, the contact force may be negative due to a large negative damping force, which is not realistic. This situation can be avoided by using the indentation damping exponent greater than one.

δ& δ 1m 2m

3m



13.6. BELT DRIVE SYSTEM

An automotive belt drive system is used for the simulation of HAT in order to test numerically. This system is consisted of 5 pulleys and a belt system. A continuous belt system can be modeled using series of a single body that has six degrees of freedom and has a matrix (6x6) force element to connect the belt bodies. Contact forces between the belt and pulleys are defined clearly.

Disturbance roller

Drive

Pulley

HAT

Sensing belt tension

Belt

Figure 7. Belt drive system

As shown in Figure 7, there are a drive pulley, a disturbance roller, four idle pulleys, and an idle roller equipped with HAT.

13-13

13.7. NUMBERICAL RESULTS

The hydraulic auto tensioner must be a reciprocating hydraulic device that can respond to frequencies up to 300Hz. When the reciprocating load is applied to plunger with 300Hz, Figure 8 shows the result of the pressure in chamber and Figure 9 shows the result the displacement of the check ball. The numerical results show that the proposed modeling of HAT is acting to the reciprocating load with 300Hz. As the load increases, the check valve closes and the oil flows out only through the leak. As the load decrease, the check valve opens and the oil flows in through the check valve.

Figure 8. Pressure in chamber [300Hz]



Figure 9. Displacement of check ball [300Hz]

The proposed modeling method of hydraulic auto tensioner is applied for the

belt drive system as shown in Figure 7. The drive pulley rotates with 100 rpm. As the disturbance roller increases the belt length, the belt tension around HAT increases such as shown in Figure 10. Due to the belt tension increases, the pressure in chamber arises as shown in Figure 11 and the oil flows out through the leak as shown in Figure 12. As a result, the plunger is pushed back and the belt tension decreases. Figure 10 shows less increase of tension of the belt with HAT comparing to without it.

13-15

Figure 10. Tension

Figure 11. Chamber pressure



Figure 12. Oil flow rate through leak As the disturbance roller decreases the belt length, the tension around HAT

decreases as shown in Figure 13. Due to the belt tension decreases, the pressure in chamber decreases as illustrated in Figure 14. Figure 15 shows the oil flow rate through the check valve. As a result, the plunger is pushed to the direction for increasing the tension by the plunger spring, and therefore the tension increases. The tension drop can be quickly recovered with proposed HAT element as shown in Figure 13.

13-17

Figure 13. Tension

Figure 14. Chamber Pressure



Figure 15. Oil flow rate through check valve

13-19

13. 8. CONCLUSIONS

In this investigation, in order to design automotive power transmitting system at early design stage, modeling and simulation methods of HAT, which is necessary component for the tension adjusting system, are presented. The multibody simulation model is proposed using three rigid bodies, which are plunger, check ball and cylinder. The plunger and the cylinder bodies can be connected by constraints and mechanical force elements. The plunger and the cylinder are interacted by hydraulic force and spring force. The forces between plunger and check ball are modeled by contact, hydraulic, and spring forces. The circle to curve contact analysis is employed for the plunger and the check ball contact efficiently. The differential equations of motion of the components and the hydraulic force equations are developed in this investigation. It can be assured that the proposed HAT model is able to respond to frequencies up to 300Hz. The proposed methods of HAT are simulated in different ways, component level simulation with reciprocating forces, and with automotive belt system. Both numerical results show reasonable responses as expected. Though it is necessary to be correlated by experimental results. Therefore the proposed numerical method of HAT shows the possibility of simulation for automotive power transmitting system, which has been challenging works for long period.



REFERENCES 1. http://www.ntn.co.jp/english/corp/news/news/20011001_2.html

2. NTN TECHNICAL REVIEW No. 61

3. NTN TECHNICAL REVIEW No. 67

4. Frank M. White, "Fluid Mechanics", 5th edition, McGraw-Hill International

Editions, 1999.

5. E. Sonntag , Richard, Claus Borgna, kke, and Gordon J. Van Wylen,

"Fundamentals of Thermodynamics", 5th Edition, John Wiley & Sons, Inc., 1998.

6. B. O. Roh, H. S. Anm, D. S. Bae, H. J. Cho, H. K. Sung, "A Relative Contact

Formulation for Multibody System Dynamics", KSME International Journal, Vol.

14, No. 12, pp. 1328-1336, 2000.

7. www.fev.com

14

DYNAMIC ANALYSIS OF CONTACTING SPUR GEAR PAIR FOR FAST SYSTEM SIMULATION

14.1. INTRODUCTION

All Geared systems are commonly used in many mechanical power transmitting systems, such as robot manipulator, automotive transmissions, etc., so as to transmit motion and power from one shaft to another. One of important factors in the gear design is the dynamic transmission error, which gear vibration, noise and other performance can be predicted by. When two mating gear is operated, the dynamic transmission error is generated by gear dynamic forces. These forces are caused by contact between meshing teeth. In other words, contact mechanics between meshing teeth, considering backlash and tooth geometric profile, is very important in the dynamic analysis of geared systems. A lot of numerical and experimental works have been published about their dynamic analysis. One of main topics in these studies is the conventional finite element analysis. Traditional finite element methods are effective for calculating quantities such as mesh stiffness, tooth deformations, and stress distributions under static conditions. But it requires refined meshes to represent the tooth contact and precise tooth surface shape for gear mechanics. Also, it takes amazingly long time to analyze the dynamics effects of contacting gears. Moreover, it is not suitable for analysis of entire system with the sets of gear pairs as well as other components [1, 2]. Another topic is that concerning the single degree of freedom(sdof) models of a pair of gears. It is because sdof model can give relatively accurate results and computational efficiency despite its simplicity. The sdof model approach in terms of entire system dynamic analysis with gear pairs is desirable from research and design perspectives. In sdof model, primitive approach is to model gear pairs with simple constraint or force element using speed ratio, pressure angle and rotational angles. Gear



systems can be analyzed with fast computational time, but detailed inputs such as tooth profiles and distance between gears are not considered directly because it is not real gear teeth contact. More advanced approach is considered by contact between teeth profile of gears. It enables designer not only to obtain gear contact position and force exactly but also to simulate with entire system in various operating conditions [3, 4].

A review of the mathematical models used in gear dynamics was given by Ozguven and Houser [5], and T. Shing et al. presented an improved model for the dynamics of spur gear systems with backlash consideration [6]. The torsional vibration behavior was investigated experimentally by Kahraman and Blankenship [7, 8, 9]. In the recent studies, a sdof model was proposed, which considers a time-varying stiffness and backlash of the meshing tooth pairs with similar formulations. However, most gear models in these numerical investigations have been used the kinematic relations between the rotational angles of each gear. It is not real contact model between bodies and needs some limitation that gear shafts have no translational displacement.

The main purpose of this paper is to develop efficient contact algorithm between meshing teeth in geared system for better understanding of the dynamic behavior of entire system. Externally specified dynamic forces, or assumptions about modeling the mesh forces by time-varying stiffness and static transmission error are not required since dynamic mesh forces are obtained by contact analysis at each time step. A simple spur gear pair modeled by using proposed methods is compared and verified with the measurement results represented by reference [7]. The dynamic modeling techniques are suggested and efficient & fast dynamic analysis of a set of complex geared mechanical system is presented in this investigation.

14.2. TOOTH PROFILE OF SPUR GEAR

The gear teeth profile is usually defined a special profile called an involute curve for constant speed ratio. However, it is not efficient to use the exact involute profile in the contact search algorithm because of its complexity of contact search kinematics. In order to approximate the involute profiles, biarc

14-3

curve fitting method which is proposed by Bolton[11] is employed in this investigation. The optimum biarc curve passing through a given set of points along involute curve can be determined by this approximation technique. The more arcs are used to describe the involute profiles, the less numerical error is occurred in approximation, but the more calculation time will be required for contact search of tooth profiles. Consequently, the real geometry of involute tooth profiles in this investigation is represented by 5 arcs with different radii as shown in Figure 1, since the error is acceptably small.

Fig. 1 Involute curves by 5 arcs

Arc segment Absolute error (mm) Relative error (%)

1 0.000229 0.00147

2 0.000349 0.00152

3 0.000388 0.00165

4 0.000409 0.00168

5 0.000461 0.00182

Table 1. Absolute and relative error



Table 1 shows the difference between exact involute curve and approximated

arc segment in spur gear with 24 teeth, 2 mm module, and 20° pressure angle. Absolute error is an average distance between points on exact involute curve and points on arc segment from gear center. Relative error is an average difference percentage that absolute error is divided by average distance of points on involute curve from gear center. Since the main purpose of the research is to understand the dynamic behaviors of system with the gear pairs, these kinematic errors might affect very small for the highly oscillating nonlinear dynamics of gear system, accordingly.

14.3. EFFICIENT CONTACT SEARCH ALGORITHM AND CONTACT FORCE MODEL

The contact algorithms for a gear pair are investigated in this section. The contact positions and penetrated values are defined from the kinematics of components in searching routines. Thereafter, a concentrated contact force is generated at the contacted position of the contact surface of the bodies. A detailed discussion on the formulation of the contact collision is represented in this section.

14.3.1. ARC-ARC CONTACT

Since the radius and angle of each arc are given at geometry, the contact kinematics between arcs can be calculated by contact logic. A marker is attached at the center of the arc and X axis is fixed to the starting point of arc. The monitoring vector between arc centers can be easily detected whether they are in contact boundary or not using the arc angles with respect to the X axis of the marker. If the vector is in contact boundary and the length between the centers of arcs is less than the sum of the radii of arcs, they are considered as contact candidate.

14-5

X

Y

Z

iX

iY

iZ

itX

itY

itZ

iR

iu

jpX

jY

jZ

jR

jpup

t

ijku

jθj

pY

jXiθ

Pinion tooth coordinate system

Pinion coordinate system

Gear coordinate system

tGear tooth

coordinate system

Global coordinatesystem

Fig. 2 Arc-arc contact kinematics

The contact conditions between the gear tooth convex arc segment and the pinion tooth convex arc segment can be determined as follows. A coordinate system and is attached to each arc origin coordinate system

shown in Fig. 2. The surface of the gear tooth arc segment is approximated by plane surfaces and the axis of each surface coordinate system is assumed to be directed to the starting arc point from arc origin. The surface of pinion tooth arc segment is approximated by plane surfaces and the axis of each arc origin

coordinate system is assumed to be directed to the starting arc point from arc origin. The orientation of the gear tooth arc coordinate system with respect to the global system is defined by

it

it

it ZYX j

pj

pjp ZYX

itX

jpX

k

ik

iit AAA = (1)

where is the transformation matrix that defines the orientation of the iA



coordinate system of the gear and is the transformation matrix that

defines the orientation of the gear tooth arc coordinate system with respect to the gear coordinate system. The orientation of the pinion tooth arc l coordinate system with respect to the global system is defined by

i

ljA

ikA

k it

it

it ZYX

jjlA

jpX

A+

i

+

r−

jZ

jj

p AA = (2)

where is the transformation matrix that defines the orientation of the coordinate system of the pinion

jA and is the transformation matrix that

defines the orientation of the pinion tooth arc coordinate system

with respect to the pinion coordinate system.

l jp

jp ZY

The global position vector of the center of the gear arc segment, denoted as point , is defined as t

it

iiit uRr = (3)

where iR is the global position vector of the origin of the gear and itu is

the position vector of arc center point with respect to the origin of the gear coordinate system .

tiii ZYX

The global position vector of the center of the pinion arc segment, denoted as point p , can be defined as

jp

jjjp uARr = (4)

arc center point defined in the pinion coordinate system . p jjYX

The position vector of the center of the arc of pinion with respect to the origin of the gear tooth arc can be defined in the global coordinate system as

it

jp

ijk ru = (5)

The components of the vector with respect to the gear and pinion tooth ij

ku

14-7

coordinate system are determined, respectively, as

[ ] ijk

Tit

Tiijz

iijy

iijx

iij uuu uAu == ,,,,

(6)

[ ] ijk

Tjp

Tjjiz

jjiy

jjix

jji uuu uAu −== ,,,,

(7)

Necessary but not sufficient conditions for the contact to be occurred between the gear and pinion arc segment are

ptiij

yiij

x rruu +≤+ 2,2, )()( (8)

pt

ijzpt wwuww +≤≤−− (9)

where and r are the radius of the gear and pinion arc segment respectively,

is half width of the gear tooth and is half width of the pinion tooth. tr p

tw pw


),(atan2 ,, iijx

iijym uu=θ , ),(atan2 ,, jji

xjji

yn uu=θ (10)

km θθ ≤≤0 , (11) ln θθ ≤≤0

where and are the angle of with respect to the gear and pinion

tooth arc segment coordinate system and and are the angle of gear and pinion arc segment, respectively.

mθ nθijku

θk lθ


22 )()( ijy

ijxpt

ij uurr +−+=δ (12)



14.3.2. ARC-POINT CONTACT The arc-point contact conditions between the gear and the pinion can be

determined. A coordinate system is located at the center point of the gear arc surfaces.

it

it

it ZYX

The position vector of the point p of pinion j with respect to the center point of the gear tooth arc is defined in the global coordinate system such as in Eqs. (5) and (6).

Necessary but not sufficient conditions for the contact to occur between the pinion point and the gear tooth are k

ruu ijy

ijx ≤+ 22 )()(

(13)

ptijzpt wwuww +≤≤−− (14)

where r is the radius of the gear arc segment, is half width of the gear

tooth and is half width of the pinion tooth. tw

pw


),(atan2 ij

xijym uu=θ (15)

km θθ ≤≤0 (16)

where is the angle of with respect to the gear arc segment coordinate

system and is the angle of arc segment. mθ

ijku

kθ


22 )()( ijy

ijx

ij uur +−=δ (17)

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14.3.3. CONTACT FORCE MODEL In the field of multi-body dynamics, one of the most popular approximations

of the dynamic behavior of a contact pair has been that one body penetrates into the other body with a velocity on a contact point, thereafter the compliant normal and friction forces are generated between a contact pair. In this compliant contact force model, a contact normal force can be defined as an equation of the penetration, which yields

32

1 mmmn δδ

δδckδf &&

&+=

(18)

where k and c are the spring and damping coefficients which are determined, respectively and the is time differentiation of penetrated value δ& δ . The exponents and generates a non-linear contact force and the exponent

yields an indentation damping effect. When the penetration is very small, the contact force may be negative due to a negative damping force, which is not realistic. This situation can be overcome by using the indentation damping exponent greater than one. The friction force is obtained by

1m 2m

3m

nf ff µ= (19)

where is the friction coefficient and its sign and magnitude can be determined from the relative velocity of the pair on contact position.

µ

14.4. KINEMATICS AND EQUATION OF MOTION FOR SYSTEM DYNAMICS USING THE RECURSIVE FORMULAS

Recursive formulas using relative coordinates are very useful for gear system

dynamic analysis since gears in geared systems are usually rotated to one axis



direction. This section presents the relative coordinate kinematics for a contact pair as well as for joints connecting two bodies.

Translational and angular velocities of the body coordinate system with respect to the global coordinate system are respectively defined as

wr&

(20)

Their corresponding quantities with respect to the body coordinate system are defined as

=

wArAY T

T &

(21)

where is the combined velocity of the translation and rotation. The recursive velocity and virtual relationship for a pair of contiguous bodies are obtained in [16] as

Y

1)i(i1)i2(i1)(i1)i1(ii −−−− += qBYBY & (22)

where denotes the relative coordinate vector. It is important to note that

matrices and are only functions of the . Similarly, the

recursive virtual displacement relationship is obtained as follows

1)i(i−q

(iB 1)i1− 1)i2(i−B 1)i(i−q

1)i(i1)i2(i1)(i1)i1(iiδ −−−− += δqBδZBZ (23)

If the recursive formula in Eq. (22) is respectively applied to all joints, the

following relationship between the Cartesian and relative generalized velocities can be obtained:

qBY &= (24)

where is the collection of coefficients of the and B 1)i(i−q&

14-11

[ ]T 1ncTT

2T

1T0 ×= nY,,Y,Y,YY K (25)

[ T

1nrT

)1(T12

T01

T0 ×−= nnq,,q,q,Yq &K&&& ] (26)

where nc and nr denote the number of the Cartesian and relative coordinates, respectively. Since in Eq. (24) is an arbitrary vector in q& nrR , Eqs. (22) and (24), which are computationally equivalent, are actually valid for any vector

such that nrRx ∈&

xBX &= (27)

and

1)i-(i1)i2-(i1)-(i1)i1-(ii xBXBX += (28)

where is the resulting vector of multiplication of and . As a

result, transformation of into is actually calculated by recursively applying Eq. (28) to achieve computational efficiency in this research. Inversely, it is often necessary to transform a vector in

ncRX∈ B

G

x

nc

nrRx∈ ncRBx∈

R into

a new vector in GBT=g nrR . Such a transformation can be found in the generalized force computation in the joint space with a known force in the Cartesian space. The virtual work done by a Cartesian force Q is obtained as follows.

ncR∈

QZW Τδδ = (29)

where must be kinematically admissible for all joints in a system. Substitution of

ZδqBZ δδ = into Eq. (29) yields

*TTT δδδ QqQBqW == (30)



where . QBQ T* ≡

The equations of motion for constrained systems have been obtained as follows.

0)QλΦYMBF Τ

ΖT =−+= &( (31)

where the λ is the Lagrange multiplier vector for cut joints [17] in and

represents the position level constraint vector in

mRΦ mR . The and Q are the mass matrix and force vector in the Cartesian space including the contact forces, respectively.

M

14.5. NUMERICAL RESULTS A spur gear pair system is analyzed for the sake of numerical verification of

proposed methods as shown in Fig. 3. The shafts of the two gears are assumed to be rigid and the only the compliance of contact force between meshing teeth is considered in this model. The gear pare model is composed of 2 spur gears, 2 revolute joints, and a gear contact element. Rotational dampers are used for resistance torque at revolute joints. A gear is driven by steady torque of 10 Nm.

11 , rθ 22 , rθ

Revolute joint & Rotational springdamper

Applied torque

Contact element

Fig. 3 Gear pair model

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Gear/Pinion

Module 3 mm Number of teeth 50

Pressure angle 20 ° Radius of pitch circle 75 mm

Radius of outside circle 78 mm Radius of base circle 70.477 mm Radius of root circle 71.25 mm

Tooth width 20 mm Elasticity modulus 29 /10200 mN×

Density 33 /1085.7 mkg× Center distance 150 mm

Table 2 Design parameters of gear and pinion

Table 2 shows design parameters of the spur gear sets which are the inputs of

numerical simulation. Dynamic analysis of a spur gear pair is simulated during 0.08 sec. Gear speed is increased up to 500 rad/sec (4800 rpm) almost linearly as shown in Fig. 4(a). It is found that the CPU simulation time is just 15 sec on a Pentium IV 3.0 GHz platform personal computer. Figure 4(b) demonstrates the dynamic transmission error (DTE= ) with respect to time domain when a gear is driven at the constant torque of 10 Nm. As rotating speed of gear is increased, dynamic transmission error (DTE) is changed by gear teeth contact. Figure 5(a) and 5(b) show the time-domain DTE around mesh frequency of 1900 Hz and 3000 Hz. Magnitude and waveform of DTE are different in each mesh frequency. Magnitude of DTE is around 30 and 3 micro meter, respectively. These results show similar magnitude and exact dynamic pattern as compared to experimental measurement results (in the reference Fig. 6 and 7) introduced by Blankenship and Kahraman [7]. The minor differences between the proposed method and referenced [7] might be expected from the dimensions, measurement settings and noises.

2211 θθ rr +



(a) Rotational velocity of driven gear

(b) Oscillating DTE with respect to time

Fig. 4 Rotational velocity and DTE

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(a) DTE at the mesh frequency of 1900Hz

(b) DTE at the mesh frequency of 3000Hz

Fig. 5 Oscillating DTE time history



The key advantage of the proposed method is the fast & efficient system simulation of geared multibody dynamic system without losing the system dynamic characteristics caused by gear pair contacts and their flexibility. An Engine system with multi gear sets is illustrated as another geared system example model. The system has 4 degrees of freedom, which has 13 bodies, 6 revolute joints, one translational joint, 14 fixed joints, and 2 sets of contacting spur gear pairs. Crankshaft in this model is rotated by gas force and gear sets are driven by rotation of crankshaft as shown in Fig. 6. In order to examine the effect of gear contact dynamics, the proposed gear contact force model is compared by constraint coupler model which should be ideal solution but not realistic. Figure 7 shows well the difference of output velocity from the final gear between proposed method and conventional dynamic anaysis using constraint only. Dynamic analyses of both models are performed for 0.01 sec. It is found that the CPU simulation time is just 85 sec for the proposed method on a Pentium IV 3.0 GHz platform personal computer.

Fig. 6 Engine model with multi gear set

14-17

Fig. 7 Rotational velocity in output gear 14.6. CONCLUSION

This research proposes an efficient implementation algorithm of spur gear

contact mechanisms for the fast system dynamic analysis. Externally specified dynamic forces, or assumptions about modeling the mesh forces by time-varying stiffness and static transmission error are not required since dynamic mesh forces are obtained by contact analysis directly at each time step. Arc-Arc and arc-point kinematic interactions are presented and a compliant force model is used in this investigation. The relative coordinate formulation is employed to generate the equations of motion. Two numerical examples, a simple spur gear pair and an engine transmission system, are illustrated and simulated numerically in this investigation. A simple spur gear pair model shows the validation of the proposed method with measurement results illustrated by reference, and engine transmission system shows the advantages of the proposed method, respectively. Consequently it is possible to simulate the entire geared system dynamic analysis without losing its important dynamic characteristics, such as vibration and noise, etc., with reasonable CPU time as represented in this investigation.



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4. Amabili, M. and Rivola, A., ''Dynamic Analysis of Spur Gear Pairs: Steady-State

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Dynamics – a Review'', Journal of Sound and Vibration, Vol.121, pp. 383-411,

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